I can't find any specs for the newer Dillon motors, but the original GE ones used on the M134 pulls around 60 amps on 24-28V @ 3000 rpm according to multiple websites. Higher rate of fire = higher power draw, and as one would expect the current spikes when firing begins.
That being said, I think the biggest issue I have with combat abstraction in general is the split of combat turns and how it affects certain things; a simple action for someone with one action phase represents a full 1.5s, while for someone with three action phases a simple action is a mere .5s. So the Vindicator spins up at different speeds for different characters. This more than any other concern is probably why I would just dismiss the need for spending a simple action to spin up the barrel; in short, I don't think it's a sufficiently balanced game mechanic.
That's not much to go off of, but i'll throw some numbers up for you:
Assuming 28VDC @ 60 rated load, 100% eff motor and 0.0 resistive load: generates 2.25 hp of work at 60hz. To go from 0 to 3000 rpm would take 11 time intervals, for a total time of 0.25 seconds.
Now the problems there are: No motor is 100% efficient. You always lose energy, usually in the form of heat. The other problem is, there is no such thing as an unresistive load. Electricity is the flow of electrons, for something to flow, it must overcome inertia to start (resistance!), then the resistance of all the other particles pressing against it (you get the idea). And so on..
Once you factor in the resistance and efficiency, the time to full rpm could be as high as 1.95 seconds (and this is assuming GE kept to its general standards as a minimum)
However, personally i can see it being closer to .4 to .6 of a second. (This is assuming the use of some complex and expensive electrical eqiupment normally only seen used on massive 2500hp+ motors.)
But that isn't a game issue. Considering the massive nerf pounding Hvar/mini guns have gotten in 5th, they do need some fixing, if only to maintain their "cool" factor.
I have always thought of these weapons as the ultimate squad support weapon. Not because of damage, but because they can totally suppress large areas of badguys easily. But in 5e they have lost that edge. There is now no reason to take a minigun or Hvar, as any assault rifle can do just as good a job.