I rolled the dice, and won, on early-September stays in London in 2017 and 2018. As a solo traveler, I found a large difference in rates between hotels with and without a/c, so I took a chance. The weather during both stays was not hot at all. This year I'll be in London in August, and that was too risky for me, so I went with an air-conditioned Premier Inn. I'm fortunate to have a travel mate this year to split the cost.
The chance of high temps in September is not great, but it could conceivably happen, so you need to consider how miserable you would be without a/c if your luck is bad. I'd worry about it more if I had a long stay rather than a short one, because I find I can stand anything for a day or two, but trying to sleep in a really hot room night after night is miserable.
You can go to timeanddate.com and check actual day-by-day weather stats for London in Sept 2018, Sept 2017, Sept 2018, etc. I'd take a look at the last 5 years at least.
Definitely pack earplugs. If you need to sleep with an open window for ventilation, you may have a noise problem.