Am I right in thinking that when I tag the chages are only kept in the MC database and not actually written to the files?
I'm only an amateur but the answer to this question is no, this is not correct. If you update the tag fields, the changes should take place within the files themselves. You can easily find out whether the tags have successfully been changed: with any album highlighted, press ALT + ENTER to view its tag fields. Then you will know whether they have been changed or not. I suggest this would be the first step towards solving your problem.
If the tag fields haven't been changed, my next question would be what method you used when you tried to change them. And if the answer is Mp3tag, there's your answer, because for some reason Mp3tag won't overwrite the Album Artist field in MC. It will write it in the first place, but it won't overwrite it once it's there. Please don't ask me why!
Also, do you have a second set of music files on the second PC? Because if you do, then obviously you will have to update the tags on those files as well. I use SyncToy to keep music files on my other PCs in sync with those on my main PC.