Ok, I just noticed that after I installed the Secret Files update, the option to play the bonus levels is now gone from the episode select screen.

Is there some kind of secret now to access this, or is it just that the game wasn't programmed to allow an extra episode so the secret files replaced the bonus stages?

If memory serves me right, the episode is still accessible, just not visible because it's placed too low down the list. Try to press down at the last episode and see if it pops up.

If that doesn't fix it, I recommend you download the JJ2+ mod from either GOG Galaxy or get.jj2.plus to fix this bug.

What bonus stages do you mean exactly? There were no bonus stages in JJ2 like in JJ1.
As far as I can see, you can play all of the episodes from JJ2 in the The Secret Files, including the shareware episode and home-cooked/user-made levels.