@alwaysalready Thank you so much!!!!!
Your solution sorted everything. I so appreciate it!!!!!!!!!!!!!!!
In case it is interesting to anyone, to try fix this before coming to this thread, I:
1. installed “Check Email” to see if there was a problem sending emails from my site (there wasn’t)
2. installed “WP Mail SMTP”, setting up SMTP with all its complications in case it was some sort of spam issue (it wasn’t)
3. discovered using the above that the registration emails were being bounced with the error message “Diagnostic-Code: smtp; 550 Messages should have one or no To headers, not 2.” When I looked at the headers, sure enough, there were two “To: destination-email-address” lines in the header.
4. Deactivated BuddyPress. Found that registration emails successfully went through.
5. Finally came here, having found out that BuddyPress was definitely the culprit.
All this has been quite stressful, it is the first website I’ve ever set up.