Hi Shihab,
Thanks for responding. Whilst I was successfully able to use All-in-One WP Migration to get the blog exported and then imported to the other domain, it just wouldn’t work for the other site.
I have now spoken to my hosting company (TSO Host) on Chat who have been brilliant and got it all fixed for me. Seems that when I used the MySQLDatabase Wizard to create the new database, what I entered as the database name (and then edited in the wp-config.pho) got an extra bit appended to the name!