OK, I've gone back today and ran in to some problems, I'd noticed I hadn't actually grabbed the entire chat (it was a very long chat) so was going back to get the remaining text.
I was struggling with the Ctrl A, but after a few experiments, I found that if you are clicked inside the name of the chat - i.e. top left of the chat Ctrl A does enable you to select all, and you can copy and paste. I just happened to have had my cursor there yesterday by sheer chance, and it actually works the same in the desktop version. However what's copied and pasted seems a bit haphazard on closer inspection. I've still got the bulk of the conversation, but am missing bits*, and bits of other chats are also pulled in. It's still going to be a huge time saver for me for what I need to do.
My fall back is going to be taking photos on my phone and using text recognition, which seems ridiculously complicated for something that should just be copy and paste. I feel there might by something that could be done do get the text out of the HTML, but image to text will probably be quicker for me.
*Edit - Scrolling up and down when you press Ctrl A allows you to grab different parts of the conversation. Stitching it together is still a bit of a pain but much better than 1 by 1.