We've recently had a multi split system installed as part of a large home extension. There's one outdoor unit and four indoor units. There's still a spare connector for a fifth indoor unit, which we'll get when our finances have recovered a bit.
Our system is a Mitsubishi Heavy Industries heat pump, so it's very economical to run and you don't currently pay VAT because it can also provide heat. I think that might be changing later this year.
We've now got a very large kitchen/diner/living room at the rear of the house. That room has a 6kW wall unit and it's only cost us £6.50 for the past seven days (just running in the evenings last week, but all day since Sunday).
The system was £7,500 including installation, which is obviously quite a lot, but it's been fantastic during this heatwave. I've been working from home and it's been bliss.
You can get smaller multi split systems for less. For example, this unit supports up to 3 indoor units and costs £995. You'd also need to buy the internal units. We've got the Diamond models that are the most efficient. You'd be looking at around £2300 + installation for the external unit and three internal units.
https://www.airconditioningworld.co.uk/mitsubishi-heavy-industries-scm50zs-w-5-0kw-
If you just want a single unit for a bedroom, tyou can get that for a lot less - but it's generally cheaper to buy a multi split if you need more than one internal unit.
I'd suggest giving them a call to discuss your requirements. We used a local installer that was recommended by our builder, but the website states that they can put you in touch with installers.
My main advice would be to get a heat pump model for lower running costs and no VAT. Also use something like Google AI to calculate the cooling requirements for the rooms (size of rooms, size and type of windows, insulation, wall construction type, South facing, etc). There's also a tendency for installers to use a default calculation of 150w/m² which would result in a system that's suitable for a greenhouse, but will probably be overpowered for your home.