Because really hot weather is so rare in the UK, we don't really hold with air conditioning as a concept. We sort of enjoy/embrace/suffer the sweltering temperatures inside & out knowing that it won't last. Newspapers run front page stories about heatwaves and switch suddenly to Fahrenheit (because it's way more exciting talking about temperatures approaching 100 than somewhere in the 30s); we get fans out and open windows and kind of revel in the stickiness, knowing it'll only last a few days.
Which is fine for those of us used to it, but if your first trip to London happens to coincide with one of those fairly rare heatwaves then it could be pretty miserable. Here in London we're used to sleeping with the windows open and are puzzled when Americans moan about the noise (it's London! Of course it's noisy!) or the lack of screens (it's only a few moths & bugs getting in; don't worry about it!)
So while aircon isn't always necessary, if the above sounds like your idea of hell, find yourself a purpose-built modern hotel with air conditioning. What you might have trouble finding is a "quaint" traditional budget London hotel with aircon. So the choice is yours....
(Nb: top choice for a budget hotel with aircon would be one of the newer Premier Inns - the Waterloo one is air conditioned and in a great location.)