Thanks to anyone who has commented over the past few days. (in fact, thank you to anyone who has commented here EVER). I wanted to reply to you and have realised that blogger doesn't give an easy way to reply to comments.
So I did a bit of a search and saw that a lot of people were using DISQUS. All the instructions on the DISQUS site seemed simple, so I took the plunge.
Install DISQUS - fine, now to import comments into DISQUS and sync back....well that part seems to have got halfway and stopped. All the comments have disappeared from here and don't seem to have made it into DISQUS. There is a happy little status message saying that my import has completed successfully and imported 100 comments, but the progress percentage is stating 0.0% and the comment list there has no comments in it.
I don't know whether it is a timing thing and whether it will come good on its own. I don't know whether the comments will come back if I remove DISQUS from the blog.
I will leave it for a day or so to see if it repairs itself and then try removing DISQUS.
[Edited to add] - I panicked too soon. Turns out I just had to wait and now everything is back. (and here I was not trusting the technology. I may have to hand in my geek badge!)