Trying to make a rules light combat system with no down time for a wargame

I'm not entirely sure where you're coming from, what you're trying to achieve, and what other factors weigh in on your development of your combat mechanics, so I don't know if I could help you...

However, since your forum title is about making it have less downtime, while the text talks about constraints, I'm assuming that you want to make a relatively large, complex game, that has your decisions made in rapid succession. I can provide pointers for that one, at least:

-Remove analysis paralysis: if you provide too many options to your players, they may become overwhelmed with their available choices, and take a long time to decide what to do. Reducing available options helps with this. Try to distill the game down to its core - what makes it fun to you? Then hammer that aspect of it with all your creative juices, and build from there.

-Reduce unit count: if a player has a bunch of things to keep track of, and has to move 20 units independently, each player's turn is going to be at least 10 minutes long, whether the combat is elegant or not. Perhaps reflavor your endeavor as spec-ops teams facing off, so you only have a handful of units, rather than large battles. Of course, if you're looking to make a 4-hour behemoth with large rulebooks, go right ahead, but know that it'll only be popular among a restricted circle.

Now, something that might hit on all three points (including a simple combat system, as requested): since you seem to have "squads", you could take the abstract "squad health" and make it actually apply to individual "units" (which really aren't figures on the board, just the squad marker is), so if the squad is hurt, you actually lose firepower. Perhaps a static bonus and a single attack roll per squad, with a bonus based on squad strength and other tactical factors, could provide the simplicity you're looking for? It would reduce the overall unit count, and would reduce overall production costs (and make it easier to prototype).

Then, perhaps tanks or other machines can take damage without losing as much combat efficacy? You can toy with that if you'd like.

There are 2 ways to reduce handling.

- As said above, 1 health for the entire squad. And keeping track of possible damage when the squad gets hurt.
As alternate suggestion on this: Perhaps not even reducing damage, until the squad members start dying. If health is less then the number of members, you know you have to remove members. This also reduces damage. But a simple, number times damage per squad member will do.

- Now, I have mentioned number times damage per squad member. But this can be done in several ways. You mentioned, not wanting to use multiple dice.
What if you simply use 1 die for a squad? But for each additional member, you simply modify the damage by adding 1 or 2? Depending on how much damage you where planning on.

On a side note, I don't think that 5 dice are bad. Unless you have to look up each die roll individually for effects. In that case, you are right, keep it low!