I don't insist on running radiators or even an engine. I saw the difference in price between kwh or gas and electricity and the thought occurred we could use an engine running from gas to turn a heat pump and it would cost the same to turn that heat pump as running from mains electricity plus we'd have the extra 'waste' heat from the engine that we could also put to use to heat the house.
Gas powered power stations don't use engines, they have multiple stage turbines etc and the final 'waste' heat from some such setups goes on to directly heat houses. We're not going to have a CGGT power station in the garden but we could have an engine.
Your idea about the exhaust to air heat exchanger could work, we'd need to fit ducting between the house room(s) and exhaust. As I see it the main problem with this is potential for CO poisoning if/when the exhaust heat exchanger fails. Would need to be very well insulated ducting or we'd expect to waste a lot of heat from the external ducting (presuming the engine is in the garden). Even if it were 100% insulated we'd still waste some engine exhaust heat because air returning to our heat exchanger through the return duct would be at least as hot as room temperature so our exhaust would still be at least as hot as room temperature.
The warm exhaust being wasteful is one of the reasons for the heat pump. The heat pump takes our remaining engine exhaust heat (after our heat exchangers have already extracted much of the heat) so now our exhaust temp might be colder than external temp, so now we haven't wasted any heat from the burning of the gas. The heat pump pumps that heat into the house/room. Could call this next bit the second reason for the heat pump but really it just increases the efficiency / scop of the heat pump. If heat pump scop is normally lower when external temp (evaporator) is very cold, the evaporator will be warmer if it is being heated by the exhaust, with the evaporator warmer the heat pump scop (efficiency) increases. A double whammy, which is why I said the engine and heat pump compliment each other.
The usual reason for having an engine is to produce kinetic energy from chemical/heat energy, this isn't very efficient because the engine does a better job of producing heat than kinetic energy. But we can use the kinetic energy to drive a heat pump (that's one form of heating) and also use the direct heat energy from the engine to heat the house (that's the other form of heating). Usually the engine wastes heat in 3 ways, there's the coolant to vehicle radiator way, the exhaust exit temp way, and the infrared/convected heat from all the external hot components including the exhaust... To maximise efficiency we need to minimise heat loss from all of those ways because they're each a significant part of where heat from burning fuel in an engine goes. Ideally the exhaust temp should be below atmospheric temp or some of the heat from burning the gas has gone into heating engine intake air (that will be at atmospheric temperature) and pumping the heated air out of the exhaust back to the atmosphere thus wasting it. If we didn't have the heat pump evaporator tied to the exhaust we could use a heat exchanger ducted air system but then our exhaust would still be at least as hot as room temperature. Perhaps use both a heat pump and ducted air heat exchanger on the exhaust, the heat pump would be at the final stage of exhaust before the tailpipe, but if we're using engine coolant to heat radiators and we need to keep that water system completely separate from the heat pump water system it's already a 2 stage system before we decide to fit a ducted system too and 2 stages seems already complicated enough lol.
I imagine we would need to keep the heat pump water (radiator) system separate from the engine coolant water (radiator) system? But how would adding extra heat to water circulating around the heat pump condensor affect the heat pump setup?
I've been thinking along the lines of the engine having it's own closed cooling system and water to water heat exchanger (with antifreeze coolant in the engine) but the engine and 'marine manifold' could be 'raw water' (radiator water) cooled. Probably a better idea to have radiator water (raw water) flowing directly through the engine, if radiator water didn't have antifreeze it wouldn't be much improvement to have the engine full of antifreeze because the radiator water in the heat exchanger could still freeze up anyway. I used to think it was illegal to put antifreeze in domestic radiators (because there's just a stop tap between the rad system and potable water system, stop tap opened to fill the rad system... but if the stop tap fails or is left open by mistake it could poison your house potable water)? That said I see all sorts of additives heating engineers add to radiator water and I doubt they're all safe to drink. If we could fill the domestic radiator system with antifreeze coolant all the better and I had no need to be concerned about the engine being off during winter while the houseowner is away on a skiing holiday with the heating system turned off, antifreeze in the common radiator and engine coolant system would prevent the engine, exhaust heat exchangers and even the domestic radiator system and plumbing from freeze damage.
I suspect whether or not any such system would be viable would be down to the devil in the detail, not least the gas versus electricity prices and not least how heat pump scop varies over temperature. Another factor which you've' said doesn't really apply to the air blowing systems you fit as much to the air to water heat pump radiator systems that are still more often seen that the type you fit (and I do take your point on that) - Scop isn't the be all and end all of air to water radiator type heat pump setups, there's also the maximum radiator temperature that can be achieved for a given external temperature? If it's -5C outside and you can only achieve +15C inside because your radiators don't get hot enough it won't be much conciliation that you're achieving that 15C indoors temp efficiently and cheaply, so you might turn on a couple of hotwire electric heaters or run a couple of indoor gas fires anyway... But sticking the heat pump evaporator sink onto the exhaust of an engine can help to increase the maximum radiator temp that the heat pump can achieve? And you can achieve 90C radiator temp anyway even without the heat pump if radiators are plumbed to an engine cooling system? Like I said early on in this thread I don't expect anything I've suggested here to become main stream, just a niche, maybe the niche includes people who already have air to water (radiator) heat pump setups fitted but do need to use other forms of heating sometimes, for these people the addition of the engine and change of heat pump evaporator might seem a good option that would come into it's own especially on particularly cold days. Also not everybody has a suitable roof for solar panels.
At the throwing ideas around stage and welcome discussion on pros and cons of various ideas, not trying to defend any specific ideas.
Do we at least agree on the following... 1. If gas is 1/3rd the price of electricity and an engine is 33% efficient at turning gas kw into kinetic rotational kw it would cost the same to run a heat pump powered by an engine that is fuelled by gas as running the heat pump from mains electricity? 2. Then we have the heat produced by the engine and we can think of ways to use that heat to also heat the house? 3. The heat we get from (1) plus the heat we get from (2) is greater than the heat we would get from (1) alone and greater than the heat we would get just by running the heat pump from mains electricity, thus (1) + (2) works out cheaper to heat the house than running a heat pump from mains electricity plus could heat the house quicker plus could always heat the house to the desired temp even if the heat pump alone could not?
Maybe my point 4 will be more contentious.. 4. In cold weather conditions sinking a heat pump evaporator to an engine exhaust (which sees the evaporator far warmer than it would be if sinked to atmospheric air) increases both the scop (efficiency) of the heat pump and the heat pump's output (condenser) temperature? The scop is not linear with temperature, there is a sharp drop in scop when external temp drops to the kind of temperatures we often see in UK winters, if we increase evaporator temp by just a few degrees we get a response in scop / efficiency that more than compensates for the energy we put into heating the evaporator, especially in terms of money if we'd usually be adding extra heat to the evaporator by using electrical hot wire heating but instead we're effectively using gas heating at 1/3rd the price because the evaporator is heated by engine exhaust temp that is heated by gas?
The graph below is something I just pasted from the internet, I don't know how accurate this is for actual heat pumps but it shows cop changing for given input and output temperatures. I say cop isn't linear for input temp because the graph curves upwards for increasing input temp and there is a stark upward turn for external temps at around 0C. Which is why I say in (4) that increasing evaporator input temp by a little can lead to an increase in cop that more than compensates for the energy we put into heating the evaporator in cold conditions (scop is specific cop, at specific temperatures). But if we would usually (without the gas / engine heating the evaporator) increase the evaporator in cold conditions using electrical hot wire energy, since gas heat energy is 1/3rd the price of electrical hot wire heat we'd save money effectively using the gas to heat the evaporator instead of electricity.

- heat pump cop vs temperature example graph.jpg (64.15 KiB) Viewed 3986 times
I didn't pick and choose the graph above, it was the first that a Google search returned. Edit, the second graph that Google returned is below, it suggests that heat pump cop doesn't reach 300% until external temp is as high as around 12degC, which to my mind with gas price being 1/3rd electricity price means if we would be paying for electricity from the grid to run a heat pump the heat pump would cost more to heat a house than a gas boiler if external temp is below 12degC. Don't know what other people do but we don't even turn the heating on for around 6 months of the year, which means we run heating for around 6 months, and during around half of those 6 months so 3 months daylight hours are very limited (so solar panels probably not much help with heating bills if running a heat pump) and external temps are likely to be consistently below 12C. The graph below is also curved, every one I've seen is curved. The problem with the rising rate of efficiency versus input temp is that when the heat pumps would be most efficient is when we least need to use them and when they are least efficient is when we most need to use them, the curve in efficiency versus temperature exaggerates that aspect/problem. Some heat pump setups do have hotwire electric heaters that heat the evaporator to shove up the scop and the output temperature, it would be more economical to heat the evaporator by burning gas than to use electric hotwire since the hotwire always has cop of 1, close to gas cop which without condensing feature is slightly below 1 but still very close to 1, condensing pushes gas cop even closer to 1, but gas is 1/3rd price of grid electricity by kwh so if cop for electricity and cop for gas are both 1 gas is 1/3rd price of electricity. My suggestion of heat pump on engine exhaust seeing low final exhaust temp would see water in exhaust gas condense.

- Researchgate heat pump scop graph.jpg (92.94 KiB) Viewed 3972 times
If everyone in the house goes out to work the same 8 or 9 hours including commuting and you all sleep for 8 hours you might only need normal room temp during around 8 hours per working day but if you work from home you might want the house to be at normal room temp for 16 hours per day. Any day everyone in the house isn't going out, like maybe Sat and Sunday, you might want the house to be at normal heated room temp anyway. Not sure how solar energy (if you have solar) would fare in contributing to running a heat pump but I might guess that if we're talking the 4 room bungalow (small house) that Gilbert mentioned and if we're talking the kwh of solar generation for 5th and 6th Jan that Rob pointed to (generated by panels on a barn roof, decent solar setup(?) though most 4 room bungalows don't feature a barn roof) of 5 or 6 kwh per day I wouldn't think the solar panels would run a heat pump without importing electricity from the grid over the necessary hours per day - If each of the 4 rooms of the little bungalow need 1kw of heat the heat pump needs to output 4kw of heat, if the heat pump has cop of 4 it needs to draw 1kw of power, the 5 or 6kwh would run the heat pump at 1kw for 5 or 6 hours, that's if the heat pump has cop of 4 but I'd expect scop at cold external temps below 12C to be well below 4 and below 3, in which case the solar panels (with battery) would run the heat pump for less than 5 or 6 hours. That's for a 4 room bungalow with a solar setup on a barn roof, ymmv...
How much do solar panels and storage battery cost? How warm will a system based on heat pump powered by electricity with solar panels keep your house in the depths of winter and will you need to run other means of heating the house during cold times and at what cost? Instead how about my idea?