I had my Beatles albums Artist tag set to Beatles except for 2 instances where the Artist tag was set to The Beatles. I have changed the tag to Beatles but yatse still tells me that these 2 instances are The Beatles.
I have tried a full update of the OSMC library but it still remains the same.
How do I get yatse to recognise the new tag?
You misunderstood how things works.
Yatse does not read tags, it gets what Kodi sends, when Kodi is OK Yatse will be OK.
With that said since you did not provide the mandatory logs there’s not much more I can tell
I need a log containing a music sync (long press music icon in left menu for example to force it) so that I can show you the wrong data Kodi sends so you can fix Kodi side.
How long do you want me to keep the log running? It is a large collection.
Obviously the full music sync.
But as said Yatse does show what Kodi sends. I do not read tags or anything I just display what Kodi sends.
So this logs will just show you and confirm you where the issue is since you do not believe me 
But in the end it have to be fixed on Kodi side.
I offer you an apology. I never had any intention of making you think I did not believe you.
I just want to see this problem, and the missing album art problem fixed.
I should tell you that this rebuild saw the tracks listed before as The Beatles now appearing under Beatles. So this issue is fixed.
I hope that this log points to what I need to do to fix the missing album art.