Here is a thread that shows what it looks like when you finish an album. You just get a little notice like the ones you'll see in the posts. Other than that, it just remains the same album as before.
You'll need to finish all the earlier Adventures and beat the mini-games in order to finish most of the albums. The good thing is once you have finished the current Adventures for that month's updates, you automatically go back to a previous month within the game and are able to do those Adventures. Just keep playing the game (try daily) and you'll be able to catch up, eventually, depending on how far back you are.
~Vicks