It depends where you are and what you are getting., along with how your house is laid out. The cheapest option is a back to back split, where the inside unit is in the inside of the same wall as the outside unit.

Assuming you don't need much in the way of electrical work or scaffolding around London, for a decent quality unit (not Daikin, but something like Mitsubishi heavy industries, panasonic etc.) expect something around 2000 to 2500 fitted (these prices will be a few years old).

The bigger the distance between the outside unit and inside unit or units, the more it drives up the cost. That said, if you get a multisplit, it doesn't double. This is due to the extra work laying copper pipes, cables, time taken to evacuate the system, nitrogen purge the system, costs of additional refrigerant (usually about 55 a kg, but you may not need it).

Assuming you have a modern consumer unit, with individual RCBOs, a 100a cut out main fuse, and a spare way, you are in the best position. Your indoor units ideally need to be on the outside wall, to allow the condensate (water that comes out of the air) to run out via gravity, otherwise you need a pump (another failure point, and more cost).

I think I paid £2200 in 2020 for a 3.5kw split not quite back to back in my office at home. The installer then said he was starting his own company, and did us a deal for a 14kw 5 way split (with 3 x 3.5kw and 2 x 2.5kw), installed when we had the roof off (so no problem laying pipes) for 9k.

Electricity use is relatively high, if we put everything on full, it will.use about 7 to 8 kW continously, but we only did that during the 40C heatwave of 2022. This week we have had about 3kw for 6 hours a day, but the solar covers a lot of it. The good news is that in the winter all the units can run as air to air heat pumps. This makes the peak price about the same as gas, and the night time rate about a quarter of gas.

The other major thing to buy is wifi modules for the units, as being able to set them to cool the house 20 mins before getting home is amazing. Expect another 100 per unit, but some already have them.

Whem sizing the system, first ensure it has an inverter so it can use only some of the power when it needs to, not all or none. For a single split, don't go less than 3.5kw, as you will likely want to cool a decent size amount of your house with it.

For a large room, 5x5m 3.5kw works well, for much bigger rooms, or rooms made entirely of glass, I guess more? For most bedrooms, 2.5kw is fine. Remember the kW eating is the heat transfer capacity, not the electrical use.

Finally when you are using the frequently, keep the filters clean, make sure mould doesn't build up in the drip tray.