Using Cortex Prime's Challenges mechanic (introduced in Tales of Xadia) might work well here. It's not quite the typical scenario used for the mechanic, but it still ought to work. It is explicitly is round-robin, which avoids too much downtime for each player. Each character in turn creates how they are overcoming the obstacle, so in this case participating in their aspect of the heist, working to tick down the challenge's pool (and on a failure, consequences are applied to the character). Once per round, the GM also gets to either bolster the pool or have the challenge do actively something against one of the characters. If a character gets taken out, there could be all sorts of consequences, including being captured (leading to a new set of goals later on in the adventure to rescue them/break out). If the challenge pool is depleted, the characters have pulled off their heist. All in all, it allows each character to be doing their own thing, focusing on their strengths and schtick, whittling down the challenge, all with ways to inject new complications/threats, neatly take out characters without disrupting the flow, and also while ensuring there's no long 'waiting around' for certain characters to finish up their things.
(I'll also have to revisit how Leverage did it!)