Another good thread or topic!
I read all the previous posts. Enjoyed the stories and the thoughts you all shared.
My experience with this was out in pacific half way from Hawaii to California.
It was morning watch (others asleep below) and as part of diligence I looked behind the boat. I was surprised to see a very large bright green sea dragon following our boat. It was about 50 feet long! It swam with an undulating motion of its tail. It had multiple bulging eyes that surfaces as it rolled in our wake. Ugly thing!
Of course this turned out to be a large section of green fish net with a few plastic floats, it was about 50 feet long that had caught on the prop. It was very dense stuff, must have been deep gill net.
We hove to, one of the crew put on a tether and mask and jumped over to cut us free.
Lots of tools have been mentioned.
For cutting tightly wrapped lines such as a jib sheet, a hack saw would be my first choice. Long strokes make faster work. Second choice would be long serrated blade (the bread knife sound very good). straight edge divers knifes are often dull stainless or have very small serrated surface, so would not be my first choice.
For cutting nets, I would use a different tool. Small lines used in nets are best cut using heavy "scissors" which makes it easier to do the cutting with one hand. So pack a pair of specialty vey heavy duty scissors or pruning shears (as used by gardeners). Really, when under water, heavy "shears" work better than most knives and are faster on loose stuff. After getting tangled once scuba diving I dove with a pair instead of a knife.
I like the plunger idea and suction cup too.
I can see how those would be handy.
But I have another idea. If the boat does not have a plunger on board, I would pass another line under the hull and snug it close to prop, sort of like a belt around the boat. This becomes a hand hold that goes around the boat from port to starboard and makes it easier to stay in place if there is current or slight swell even if coming up for air or break or communication. Not too tight to prevent the diver from getting a grip. The diver goes up and back down holding the line. Remove the line after the diver is back aboard.
Others have mentioned some of the risks: head injury from hull coming down on diver, entanglement, hooks in fishing lines, and premature engine starting. Good to be aware of those.