In our campaign, (we're at level 19 now), we've been doing a couple of things to speed up the action:
1. Our DM tells who the next 3 players are - tells them to be ready.
2. Some of us have started pre-rolling damage. For my Avenger,
who crits a lot, this is very helpful. Right before the start of the session,
I have one other player watch as I roll out 1w, 2w, 3w hits, as well as
hits if I crit, covering my main weapon, my offhand weapon, and my
holy implement. I still do the attack roll to see if I hit, taking that out
really removes the excitement. This has helped a lot though, its gone
from the DM getting a dmg total in 5 seconds, to him rolling his eyes
and saying, "call NASA" when I critted.
In my limited experience, its cross-talk at the table among the players,
as well as a player not knowing what they are gonna do (you know
you're in for it when the DM calls their name, they look up from their
laptop or whatever and ask where they are standing), that really slows
combat down.
HTHs.