You do first need to address if this is a valid concern or if she’s just playing up. Now, I assume you’re in the US? Your contracts work differently than in my country, so:
Is it written on their contracts that OT is an expected part of the job?
If so, what does it say about required notice to inform your team of OT?
Is OT putting them over their legally advised hours under US law? Do you even have that? Have these people opted out so that they can work as much OT as they see fit?
If your office needs sudden and immediate OT then that sounds like poor planning. Of course, it can’t always be avoided, but if it’s common then where does the need come from? Do you need to get the lines of communication open with another team or manager etc?
Can you foresee the need for OT coming, and warn your team accordingly?
This woman is pushing back in an unprofessional manner. Maybe she doesn’t care. But you have to find out WHY she is pushing back – that’s your job. Call her in for a one to one. Mention the incidents of lateness and her unprofessional attitude. Ask her where this is coming from. She may have a sick family member, or an illness, maybe she’s stressed, maybe she’s just sick of cancelling her personal plans because of poorly planned OT. You don’t know! Take it from there, if there is something wrong then see if you can alter her hours for a short while, if say, she is taking a relative to an appointment before work which leaves her late. Make reasonable adjustments if necessary. Document the meeting, the concerns, and the solution. Give her a timeline in which to improve or reassess. Book another one to one.
If she’s just being rude and unprofessional for no reason then put her on performance procedure where you check in with her performance and work level.
I will say that you come off in this post as not caring one jot for your employees. Perhaps you’re just stressed and frustrated, in which case I appreciate that. Feel free to vent. If however, you just see these people as puppets there too do your bidding then no wonder you are getting push back – people can tell if they’re appreciated or not.
Sure, a job is a job and we’re there to do what we’re paid to do. But that doesn’t mean that some respect and appreciation doesn’t go a long way. You say you work twice as hard and that’s why you’re manager (congratulations on the promotion) but as a manager your job is to care for both the business needs AND your employees which is a very different ballgame.