I have been to London during a heat wave, and I can confirm what you're saying. All the restaurants had to move their tables out to the sidewalks to keep going because it was too hot inside the buildings, none of which had air conditioning.

Since the tactics of the "we're-all-gonna-die" climate doomers have remained consistent, the number of deaths is probably an exaggeration to get the most fear traction out of it for their global warming narrative (climate change narrative when it's not warm). Even so, it does encourage a bit of Yankee pride. I grew up in Florida without air conditioning in dark ages before the internet and mobile phones, and I don't remember hearing any reports of people dying. In summer, the only news was that it was very hot and would remain that way until October, when it would be less hot.