We have one move we reckon speeds things up a fair bit (and makes it more enjoyable): Purchase & repair units doesn’t need to be finalised until the end of Combat Movement.
Often, you need to put the combat movement on the board to visualise it. This makes less and less difference the more I play but I would call it invaluable in my early days of A&A. Certainly less radical than some of the other suggestions here.
We only have 3 colours of dice and red is always the highest number. Although I can certainly see the merit in 4 colours with 15+ of each - and never deviating the number a given colour is rolling is also a plus. Two trays to allow offensive and defensive rolling simultaneously is pretty good too.
Another point is that combat movement which doesn’t involve rolling dice is generally allowed to be retrospective if forgotten (not if affected by something which has happened though). NCM can be allowed too if not affected by a declared combat move. This is open to interpretation. When I’ve played Triple-A, not being able to go back (without invoking edit mode) was a different experience.