Unfortunately not. Most houses in the UK are heated using a 'central heating' loop. A closed loop of water that circulated through pipes run through the house and through large radiators in each room. A central boiler, usually a gas boiler, burns gas to heat and circulate this loop to provide whole-house heating, while also typically using a heat exchanger to heat the intake cold drinking water supply into a whole house hot drinking water. (Thus, most houses have cold, hot, and central heating piping).

If you switch out the gas boiler for a heat pump, it can still heat the hot water and heat the central heating loop. But it can't provide cooling that way. There is no infrastructure in most houses to run AC ducting or refrigerant pipes.

You might think that you could simply cool the water in the central heating loop, and therefore make all of the radiators very cold, and use that to move heat out of rooms. In theory that might work, but in most houses these central heating pipes are not insulated and run under floorboards. If you make them cold then they'll cause condensation, leading to water in all kinds of small spaces, and likely leading to warping, damage, or mould.

In the UK, retrofitting AC into an existing house is a huge undertaking in most cases.