Wouldn't it be the opposite like it was with Chelsea's ban. City would be suspended unless CAS can overturn the decision in time, right? So City should want the judgement by May or June at the latest.
I'd imagine City will hope any punishment is suspended pending CAS' findings - certainly they'd push for that. By that measure they'd want to drag this out. But they couldn't control when UEFA rendered the judgement so it makes it difficult for City to stall.
As I see it they either cooperate for a swift judgement or request for the ban to be suspended pending the appeal, and then try to drag the appeal out into next season. Can't see CAS going for it though.
EDIT: Furthermore, it's almost certain the PL wont act until CAS confirms the UEFA decision, so even if City can't overturn the judgement or punishment, if they can drag it out a bit they might be able to forestall any penalty for the domestic season.
As for the lawyers, there's only so much you can pay, even to a sports lawyer. And I'm sure there are plenty of clubs ready and willing to contribute to UEFA's funds for this.