There is some pretty compelling ideas happening in this thread. If we got a well executed set of initiative and "response" abilities I think it would create a VERY robust tactical experience without bogging down the interface or speed of combat much.
Some refinements upon above ideas;
Initiative - I'm a little uncertain whether i prefer a "defender first" or a "movement by unit initiative" approach, but in my head, the unit initiative approach adds a great deal of options. Sword-bearing units will act before hammer bearing units, Cavalry units act before non-mounted ones, units with more experience act before "greener" units, etc. Everyone gets one turn before anyone gets a second.
Alternately, a system in which units have a speed value could also be interesting. All units move on the first "turn", then act again every <Initiative Value> turns. A really fast unit might have an initiative value of <5>, which means that every 5 initiative steps, it takes an action. A unit with an initiative of <8> would only get to act every 8th step, which means that the <5> gets 3 moves for every 2 of the <8> unit.
"Overwatch" Abilities - I would be quite excited about these. Allowing units to give up their entire action in order to act responsively would be quite cool. Pikemen bracing for a cavalry charge, Archers attacking units as they move in to attack another unit in melee, swordsmen getting a "first strike" against an oncoming melee unit. These would definitely take some fidgeting to get right, and would certainly require some form of UI notification for players (so we know easily which units are braced for response). In order to make these things work right, though, you would want to have a more fluid initiative system (such as the <Initiative Value> system I discussed above so that units hold their actions, and then get their initiative order reset upon acting.
The one thing that I would REALLY emphasize as critical for the combat update is for melee damages to be calculated and applied for both sides simultaneously. That will make the combats more realistic (which is what we need for a fantasy game, right?), and make it less possible to really "game" the system, making the high defense units kill the first few models while taking low damage, and then have the high damage low armor units finish them off with no casualties. It's silly.