OP, you will need to apply for planning permission. Are you familiar with how it works? You need it even for non-listed building outside of conservation areas.
The rules in England are that any "plant", like the external unit of an aircon system, must produce a noise 10dB lower than the average background noise of the area.
The proper way to assess this is to call a noise assessment consultant who, for about £1k, will measure the noise with some kind of big, professionally calibrated microphone, then calculate the noise that your neighbours would hear, taking into account distance, walls in between, etc.
The external units tend to produce between 48-52dB of noise. The background noise in a quiet back garden tends to be 33-38dB (unless you overlook a main road, are under a flight path, etc). if you have a huge garden and a fully detached house, distance alone (from your neighbours' windows) might dampen the sound enough; if you have an average mid-sized terraced house, you will most likely need some kind of noise-dampening enclosure. These things are big and expensive
. If you have a small garden, it would take a lot of space. If you don't even have a garden, well, you cannot hang it on a wall.
All of this is just for the noise. Depending on where you live, you may be subject to other rules on the "eyesore factor".
An alternative is a water-cooled system without an external condenser
; and therefore with no need for planning permission; these systems get rid of the heat not by expelling it via an external fan, but via the water pipes.
Many people install aircon without planning permission, but it's risky
, because aircon is expensive, and you risk that a lovely neighbour might complain and get the council to force you to uninstall the whole thing.
I know of 3 companies that install water-cooled systems (you can google the names to find the links)
If you want to go for an ordinary system, you can get quotes from the first two above, or from DgAirConditioning ltd. The Air Conditioning Company is probably one of the largest in London but also fairly expensive.