This is only sort of delegating, but it is definitely a speed improvement (which I believe is the true intent of the question). Our group does what we call "Simultaneous Combat." This involves some rules changing, but the speed increase is worth it for us!
We do group initiative, which has its pros and cons, but it is necessary for the system to work. The team with the highest initiative goes first, and it cycles through teams from there (many ways to calculate the "highest initiative", but you can use the max for simplicity). If fast players feel slighted, you can give the fastest player a free "surprise round" move action or something.
Each team's turn goes something like this:
- Questions: if anyone needs to ask anything, do it now.
- Callouts: each person can say up to one sentence (ex. "Cover the flank!" or "Heal coming your way!"
- Actions: Go around the table and each player on the team announces what they are about to do ("I go attack that goblin!").
- Rolls: Everyone rolls attack and damage (or whatever). Go around the table and each player announces their rolls (something along the lines of "18 to hit for 6 damage").
- Results: The DM announces what happens. (At least in 3.5,) each round is supposed to be "6 seconds" in which everyone is acting at the same time, so if two people attack a creature, it doesn't die from one of their blows, they both kill it together.
Then any other summarizing is done.
The main improvement is that everyone does their math at once and that everyone is constantly involved. Before, everyone spent a lot more time chatting in between their turns (it could take a couple minutes to get around the circle and through the enemies). Some people had trouble refocusing ("uh... what happened since last turn?").
This took our combats down from 3 hours to about ~1.5, so it really worked for us. Let me know if you try it and how it works!