As you are having a problem with the remote key and to rule out that area of worry you could put a fresh remote battery in and reprogramme the key to try and operate remote locking using the instructions in one of the online tutorials. Google Mazda NC remote key programming for some info.
Sorry, but, as pointed out, the remote unlocking is completely separate from the immobiliser system.
JS
I read that but It is coincidental and not desirable/unexplained as to why its not working. If it is returned to working condition then you have one problem remaining and not two.
Thank you all for your advice and ideas. Problem solved! A blown main 120A fuse! This was after a second check, we didn’t spot it the first time.