Well, if the engine shuts down from heat in the tabletop game, it takes one turn, 10 seconds, to start up again.
So I´d say, from standby to combat-ready and vice versa takes 10 seconds each.
From "sitting around in the hangar" to combat-ready? Longer. Say 3 turns to strap in, another 2 turns to go from cold to standby, then a sixth turn to go from standby to combat-ready. On each turn except the sixth, make a piloting roll; on 2, you screwed up and must start over, any other failed roll means that startup does not advance this turn.
You could also, for example, rule that any weapons that are not particularly energy-intensive (say, any non-energy weapon except gauss rifles) can be fired while the Mech is powering up, but with a +2 penalty because the sensors aren´t fully online yet, targeting computers, Artemis IV etc doesn´t work and Streak launchers won´t fire because they can´t acquire a target.