I would suggest looking at two separate areas. So the practical things that might improve life a little at the moment, and then when you are having a cooler time and less stress my usual advice about the two lists pro's and con's regarding moving and the options you might have.
I have moved quite a lot in my life including living in Syria and Portugal and spending time in Sweden. So the very basic day to day stuff. Loose cotton or silk clothing is obvious , but I would also include for you no bra , whilst at home, unless you feel uncomfortable without it. Anything restricting the movement of the cooler air. Then if you have old hot waterbottle, fill with cold water and a little ice and at night before you go to bed run it over the sheets and pillows to give you a cool start , or if not , have you one of those little freezer blocks that go in lunch boxes? Wrap in a towel and use in the same way. In the day if you are sitting, you can put your bare feet on them, cooling and if you have a couple of them you can rotate them and refreeze them. If you have to go out shopping yourself, then besides going as early as you can manage, if you have a bottle of water that you have chilled in your fridge, then get a flannel and wet it with cold water, wring it out a little and then put into a plastic bag and put next to the water bottle. That way you have cool water to drink, and it keeps the flannel cool, which you can use to cool your face and hands and wrists. At home if you also in the morning wet your hair comb it slightly and leave it to dry, and keep some cold water in a handbasin , where you can put your hands in and let them get very cold , as and when you need to.
It really does work to keep windows and curtains shut all day, even though you dont always feel it is so. If you have wide windowledges then put a bowl of cold water on them , or if you have a fan going , put a bowl where the air will go across it and it helps to cool down. Might you consider shutters on the windows outside as a simpler possibility, to help with the heat.? I expect you have thought of most of these things and I am not trying to teach my grandmother to suck eggs! Have you found the coolest place in your house. One place I lived in I used to take a chair into the hall especially in the middle of the day as it was the coolest place there, and in another one I used to choose the kitchen with cool tiles and a blind to pull down to keep the sun out. You might also look at the cost of an outside blind/sunshade. If you had a roller style that you actually pulled outwards like the shop awnings, then it would not be too difficult to pull out and a thickish one will help to keep the sun off.Pull it out first thing in the morning and it will be a help all day.
When it is so hot , as they say, all bets are off! Of course I let people know, but I dont go to group meetings held in hot rooms, or places, and accept that food will be whatever I feel like eating, and least effort. Keeping off alcohol , which I think is not cooling , so with that and less food hope to have lost a bit of weight! Keep the energy for the vital or important things
Looking at the more longterm situation, I think my old way of doing things like the old game of consequences, where every time you think of something good or bad, you write them on the appropriate list and then fold them over, and add more as you think of them . Then when some time has elapsed and you are in a cooler climate and feeling more relaxed, you open your lists. Then you can see patterns in your ideas and thoughts and you might find you are surprised at some of the things you read.
So you will be looking at staying near family and all your friends and contacts you have now, and the idea that you might move back up to Lancashire. The hassle and cost of moving might be compared to the cost of perhaps making at least the ground floor of your house a/c and also if you had really good double glazing with wide gaps between the glass, that would keep down noise and be warmer in the winter too. When I was in Sweden they had both triple glaze and double glaze with a wide gap between the glass. Certainly worth having and less noise is also very good. That combined with the awning idea might make a bid difference.
I dont think "going back" to somewhere works very well. Things change over the years. It is worth going to an area you like, but then you need to look at all the ups and downs there too. One thing about London is there is good transport one way and another, and you would need to think about how you would get about, driving or cost of taxis etc. If leaving London is the main thing, might you consider somewhere further out but not so far from your family. ?
When I made my last move, I thought of all the rather negative and pessimistic things, such as what if I could no longer drive, walk upstairs etc etc, and then I was able to see what was important to have available, such as a railway station, access to a hospital etc etc. So far , fingers crossed I am still driving but very glad that I now have a ground floor flat. Perhaps when you have read lots of posts , and picked up ideas that might be useful to you, you could then try and make a rough idea of the various costs, so that you can compare them as you may find the various possibilities have differing costs. Then you might speak to your family when you have got that far and they may have other suggestions that you havent thought of. Ah well I have gone on long enough! Time for a cool water and flannel. Hope you find the best solution possible. Best wishes