You can substantially reduce your electricity bills if you have solar panels and a battery. What about if you have a heat pump? It certainly does make a difference but most of your heating demand is in the winter and most of your PV generation is in the summer. You would need an enormous battery to save enough in summer to use in winter. Suppose you put in enough solar panels to match your annual use so you are net zero for electricity consumption. What would this mean for your bills and for carbon emissions? This chart shows example savings from an energy use model.
I have used a model of a fairly typical semi-detached home. Using a combi boiler, the gas bill for this model is about 10,500 kWh/year, electricity 2,800 kWh/year. Switching to an air source heat pump for heating, electricity demand increases to 6,700 kWh/year and I have given the house solar panels to match, so net zero electricity over the year. The battery charge-discharge efficiency is 90% and the solar inverter efficiency is 95%.The weather is typical for Cambridge, with hourly resolution.
NB. Few people have more than 4 kWp, partly for lack of suitable roof area, and partly as you need special permission from the DNO (Distribution Network Operator) to exceed this. If these were all south facing you might get 3700 kWp from them. I have also modelled the impact of an array that size. The savings are smaller but definitely worth having. Also you can increase the savings with a tariff that has lower rates for off-peak electricity - I have used the Octopus Cosy tariff as an example.