Thanks for the reply. When I checked the settings mentioned they were all set. However, in the iMessage settings there is a section headed You can be reached for messages at:, which lists a few email addresses (not sure how they got there) and my mobile number. The email addresses were ticked, but the mobile number was not.
The Sync Now button is greyed out. It was open and I clicked it, but it is now greyed out, but no messages are appearing in Messages. I sent some test messages from my mobile and they only appeared on my MBP when the phone number above was ticked, so I have left that ticked.
It says it can take some time for messages to sync, but the only entry on my MBP is the 1 thread I used for the tests, and that only includes the two messages sent when the mobile number was ticked.
Will it take a long time for the history to sync onto the MBP, or is there something else at play here?
FYI, I don't think I'm bothered about the chat history, but if it needs fixing then it would be good to fix it.