Thanks for the reply. I think I might have found the cause of this.
If you go into Preferences - Tags (2), what are your artwork providers and what order are they in?
Here's what I have now: -

This is working. When I made this thread, I had Google at the top and fanart.com second. I swapped the order of those two around and now it works. I don't understand why though, but it does.
As for the display mode, these are my settings: -

Under Theater Mode I don't even see "artist picture" as an option. I have "Cine View", "Default Narrow and Wide" and "Rdio Style (embedded).
However, the setting I was using before - having "Artist picture" selected as in my screenshot above, was working fine until recently.
I wonder if there's some issue with receiving artist pictures from Google, however, I assumed that by having multiple items checked on the list, it would go through each of them in turn, but that doesn't seem to be the case, because with Google at the top, I get no pictures for Madonna and Rihanna, for example, but with fanart.com at the top, I do.