I second (or third?) the recommendation to look at heat geek.
Re the house heat-up time, how warm do your radiators feel? It may be that your system is setup for an ultra -low flow temperature. This can be extremely efficient but works best if you are keeping a constant temperature. If you were able to turn up (or get an engineer to turn up) the flow temperature your bills might rise slightly, but your system would be more responsive. This might be the simplest way to get a few hours at 21 when you want it. You could also experiment with setting the system to say 19.5 all the time (using your current flow temperature), and see if that's actually just as comfortable .
To keep the high efficiency but get a more responsive sytem, there are two bigger changes you could also consider (they would be additive so could try either or both)
These would be:
Better insulation and draughtproofing. That way the same amount of heat warms your house up faster.
Bigger radiators in the rooms you want warm - a bigger radiator gives out more heat even at the same flow temperature, again, warming the room faster. Heat pumps do work much better with larger radiators ie double or even triple thickness ones, of a good size.
This should all have been checked by the installer of course, but I realise you weren't there.
Re the hot water, again if the system isn't giving you what you need you could get a bigger tank (this is assuming the tank you have has the correct coil -again hard to know if you weren't there but there may be some info somewhere?)
Otherwise again, how hot is your unmixed hot water? Heat pumps vary in the top temperature they can reach but you may be able to turn it up, which means a tankful will go further. Once again it does cost more to heat to a higher temperature but you may feel the convenience is worth it. And it will still be cheaper than using an immersion heater.