I'm right on the edge of my knowledge base when talking about labour - so, take all of this to your midwife and ask for a second opinion!
The pudendal nerve is an interesting one, in that it has a number of twists and turns as it runs it's course. So, yep, it can get pinched in labour, and, as it supplies your urinary sphincter, then, yep, in theory, you would get pain from a compressed nerve when you pee.
It also supplies some of the muscles of the pelvic floor (the others are supplied from nerves coming from a level up or down) - so, yep, if you had pudendal entrapment that lasted over several days then it wouldn't be unexpected that you had a loss of bladder control.
Oh, and yep, it does exactly what you said - relaxes as you poo or pee. Being able to relax your pf is also important, and a too tight pf can cause pain. It's treatable, of course, but pelvic pain's not pleasant (guess you know that, mind!)
Nerves do heal, but, very slowly. So, if they are injured they take many weeks to get better. Heal at a millimetre a day, and during that time you might experience odd sensations or numbness or pain.
If you've had good function since then we can assume that any theoretical nerve damage has gone. Huzzah.
Yep, a stronger pelvic floor is associated with a shorter pushing stage and improved recovery time with a vaginal delivery. So, yep, it's got a role in labour.
Would be interested in what your MW says, and, I'd be asking her to write it all over your notes and make sure that if you do get the same problem the length of time is noted and so the health of the nerve is monitored.
Fingers crossed you get a much easier time of it this time around.