The main issue is that a heat pump takes a lot longer to heat up water on demand than a gas boiler. A gas boiler as used in the UK typically has a heat output of 25 to 50kW. Where as a heat pump for the same sized home can produce maybe 8kW of heat - given the right outside weather conditions.
The gas boiler can directly heat up cold water and provide sufficient flow for a hot shower, but the heat pump cannot. Because of this you need to buffer the heat in a hot water tank, usually something that's a few hundred litres, and heat it up over time. I believe boilers using heating oil work somewhat sim
In the UK houses heated with gas usually don't have a hot water tank, or even anywhere to put it. You can get all-in-one heat pump units that are the size of a large fridge which have a built in tank, but I've lived in places where there wouldn't even be space for that.
The boxes in the showers used in the UK are electric. They work pretty well (esp. considering most homes in the UK are single phase), but I am always concerned by the fact that an electric wire carrying 30kW of power is running right next to running water :-) Other than a gas boiler older homes may have a gas fireplace, but they are not so popular now.
There's also issues of the flow temperature for the heating system. Traditional central heating usually has a temperature of 70-80C, where as underfloor heating systems using a heat pump usually have a max of 35C. With a heat pump the efficiency drops with the temperature delta, so they are optimised to produce lower temperature water. But traditional radiators work better with higher temperatures.