I realize this is a subjective question. We use air conditioning in July here in the midwest but we don't want to be miserable in a heat wave either. Lack of air conditioning limits our options for rentals in desirable neighborhoods. We need two bedrooms and preferably two bathrooms.
IMO, no you don't need air conditioning in London in July. I've been in London several times in July and have never needed it, one day it could be warm to hot, then the next day it could be cool.
Only you can prioritize what's important as far as neighborhoods, a/c, etc. If you end up choosing an apartment without a/c and it's warmer than expected, you have some options. You can get some fans from an electronics store. Or ask the owners what happens if it's unseasonably warm. They may have fans available. Make sure there will be windows you can open.
I will defer to the locals about the normal temperatures. But I will offer that I was in London during a warm spell (low 80s) and it was very comfortable at night which is what matters the most to me. I don't sleep well if I'm hot.
It may or not be hot in the flat but I can guarantee that in July it will be hot in the Tube and air conditioning is not the norm in the Tube. Also, the new Routemaster Boris buses became christened the Roastmaster for a similar reason.
The majority of places do not have air conditioning. Last July I was in York and it was a heat wave 80 -85 degrees. Unusual and yes, when we got to our bed and breakfast and she showed me the room upstairs with a tiny window - I was in a panic. I too deal with midwest summers and I am very accustomed to consistent room temperatures. You won't get this in England if it is hot - the good thing is that you can usually open the windows (no danger of gross bugs in England coming in) and it never is as humid as the midwest summers can be. I actually declined my B&B that night and she gave us a nice drink and let us go on our way. Found a nice farm house with big windows that we stayed in - we kept the windows open all night - but we were in the Yorkshire Dales. You may not be able to open windows as much in the City - but that is what the English do when it gets a little hot. It usually doesn't stay at those temperatures for more than a few days.
I would never stay someplace in London in July that did not have air conditioning. My husband and I absolutely hate hot weather, and nothing ruins a vacation like not being able to sleep well at night. It can definitely be hot in the summer in London. It may not be, but you can't predict. Often if it's hot in the day, it will cool off at night, enough that you can open the windows and be reasonably comfortable. But the problem in a big city (depending on where you stay), is that open windows can mean a lot of noise. So it all depends on your sensitivity to heat.
My family and I spent 10 days in London towards the end of July a year and a half ago. Yes, it was warm during the day but certainly not unbearable (for us!). We stayed in a South Bank flat and with all the windows open enough to let air in, we were all very comfortable sleeping. That said, there really is no way to truly predict what this coming July's weather will be nor what each individual's tolerance for heat will be. It is definitely more humid there than where I live! As you said in your original post, subjective. Good luck with your search as places with ac are not plentiful in London and I am sorry I don't have any recommendations of places to fit your needs.
Better to have AC and not need it than the other way around.
You won't always be able to open windows, or want to. (Street noise) Buildings in large cities block breezes, too.
The "two bathrooms" might be even more of an issue than air conditioning. It's very rare for any but the largest houses in Britain to have multiple bathrooms.
Personally the last thing on my mind when going to England is air conditioning. Even if it's hot during the day it doesn't stay hot during the night, open the windows and enjoy the English air. I'm from the Midwest also and use air in July, but never considered it while in England. Your bigger problem will be finding a flat that had two bathrooms. Have a great trip.
Thanks for all the great responses.
"Personally, the last thing on my mind when going to England is air conditioning." How true!
I had to laugh when reading the post by mpaulyn from Missouri, because it brought back a memory of years past, staying in one of the old Thistle hotels without air con:
"the good thing is that you can usually open the windows (no danger of gross bugs in England coming in) "
On one of our trips to London, we had our windows open about one foot to let some air into the hotel room. On the windowsill outside the window stood three pigeons, looking longingly at the crackers we had on a small table beside the window.
No screens on the window to keep them out. They were just too shy to come in.
I do agree with what RobinZ wrote,
" I've been in London several times in July and have never needed it, one day it could be warm to hot, then the next day it could be cool."
So true. You never know what weather you're going to get in London.
We were in London last May during a heat wave. May 4 I was touring the Tower of London wearing a sleeveless blouse, capri pants, and sweating in the heat. Fortunately, we had booked a hotel with air conditioning. The heat wave continued for about a week.
Then we left London and went to Salisbury, stayed in a hotel without air conditioning. It did not matter because by then the temperature had dropped about 20 degrees, and it was cloudy, windy and slightly rainy. We went out to Stonehenge about May 14, and wore sweaters with windbreakers on top.
Went from there to Stratord-Upon-Avon, where it was sunny and warm; not hot.
Back to London for the last two weeks of May. Still warm. We checked into a hotel without air conditioning...and roasted until we could cancel and move to a different hotel. The problem with opening the windows in that hotel was all windows faced the street outside with plenty of noise all night long!
Go for a hotel or apartment with air conditioning.
If you go for one without AC, make sure you are on a side street in a very quiet neighborhood; then you could just open the windows. You don't want to be on the ground floor with open windows, for obvious reasons.
My husband and I spent all of August, 2016 in England. We were in London for 23 days and the Cotswolds for 8 nights. We stayed at the Conrad St. James, St. James Park. It is a Hilton property. There was a heatwave for 8 days. We really enjoyed going back to our hotel and enjoying the a/c. We were so tired from the heat, as it was miserable.