If it was a post about a man who had cancelled a first date because of 28 degree heat, he would be ridiculed on here.
I'm sure its possible to find a date venue thats air conditioned in the whole of London within a short journey of the OP's home - I'm assuming she won't be travelling direct from work because her condition would appear to rule out working, travelling to work, etc too. Equally, depending on circumstances, travel out to somewhere like Epping Forest and go for a walk in the shade or similar.
But I think it's probably still possible to make a relationship work with some people, if there is otherwise enough love, understanding and affection.
But this is a first date. You presumably barely know each other, never mind have love, understanding and affection. If you create a bad impression by cancelling a first date for a reason that sounds to the average person to be quite a flakey reason, you are fairly unlikely to get a second chance.
I do speak though from having stupidly put up with a flakey ex boyfriend for too long, who would cancel dates because he had a cold or because it was raining, so I may not have an entirely neutral perspective. In his case, it was all about him creating himself to be a fragile sort of victim who could never be questioned and who everyone else had to run around after to keep happy.