OK, so your nut slots are OK. The "action" usually refer to the string heights at 12th fret, which you said was 15 - 20. Mine is about 4 mm so your number sounds awfully large. That may affect the fingering in general and you still want to get checked out, but the difficulty in Bb may not be due to the action.
To barre a Bb chord, remember that you're holding all four strings but only the E and A strings matter. So you can ease up the index finger pressure on G and C strings a bit. Some people only barre the E and A strings using the index finger tip. I prefer to barre all the way acress, since that allows me to place my thumb at the center of the neck and gives me a good leverage.
No matter, I much prefer to transpose the key from these "flat" chords to D or C or G and avoid using Bb (and Eb and Ab).....
Cheers
Chief