If I wanted to customise printNestedAlbumsList() to display tags on each album, is there a way I can do this without modifying the core?

The simplest way is to make a copy of the function in your theme and rename it differently. Then customize that copy.

I want to edit what appears on admin-edit.php...

Possibly more accurately I'm referring to the output of printAlbumEditRow(). Ideally I'd like a new div in there that outputs a comma-separated list of tags?

You will have to hack the code to do that. Remember that any such changes void your warranty in as much as any updates will overwrite the files and we cannot provide support for problems the changes might cause.