Really, the having to have a cath thing is very minor.
I had this happen after DD2's birth. She was born without medications or interventions at a hospital. She ended up being posterior facing, so came out face up rather than face down, which put a lot of extra pressure on the urethra and caused more swelling than is usual.
It is important that your bladder empties because a full bladder keeps the uterus from being able to contract back into it's original position correctly. This is really important for preventing heavy, heavy bleeding. And also of course for your comfort and the prevent hydronephrosis (where urine backs up into the kidneys) from your bladder being over full.
After having her I couldn't pee. Tried the shower thing. Tried waiting. Nothing. And I was bleeding quite heavily. Like multiple pads in an hour heavy. I was really freaked over the cath, but honestly it was no big deal. A slight pinch, they drained out a couple cups, my bleeding slowed down, I took an anti-inflammatory, and by the time I had to go again it was fine.
If you're having a homebirth ask your midwife if she brings cath supplies. My Mom just had surgery & ended up having to self-cath at home for a couple days. Seems like kind of a silly thing to have to transfer to the hospital for if it can be avoided.