I believe the Benelli M1/M2's are the fastest. They are recoil-operated, so the bolt starts to move almost immediately, as opposed to gas-operated designs where the wad has to get past the gas port, then gas pressurize the piston(s) before anything starts to move.
I have some firsthand experience with this as well. When I first started working at gunshops, my coworker Zack and I did all the test-firing into a snail trap. We used to take turns and see if either of us could "out-run" a gun cycling and end up with a dead trigger or jam. (We did this after firing 3 slow rounds to check normal function). We were able to outrun many shotgun models, including Remingtons and Berettas. We never could outrun an M1 or M2. I recall one evening we both tried quite hard and just couldn't do it. We never had the chance to try an M4, but they're gas operated, so I can't imagine it being as fast. Watch some Tom Knapp videos in slow motion. It's extremely impressive (both the gun and the shooter).