The role has a start time and a finish time. Those are defined in the contract of employment. They arrive in time to start at the start time, unless it has been agreed in advance that they will arrive late.
>she has already come in late/left early multiple times
Why? If you have finished work and decide that she can leave work early, that's fine but otherwise she is there until the agreed finish time.
>+ some flex as needed
How have you defined that in your discussions with your nanny? To me that means that on occasion a later finish time may be needed. In my case, MumBoss or DadBoss may get delayed on their way home, so if MB is stuck in traffic then I cannot leave until she gets home. If DB is working from home and has a phone call come in close to finish time, he can't just hang up the phone at 5pm... it may be a little bit longer before he is able to take over.
If your nanny needs to come in late for some reason, then you may be flexible on that occasionally. However it's been two weeks and it's been more than once, and it may not have been agreed in advance. It sounds like poor timekeeping to me. You need to set the expectation that they arrive at work on time, ideally a little early.
>occasionally steps out for lunch while the baby naps (but baby inevitably wakes up before she is back).
That is not part of the deal surely. She is paid for the entire day, does not get a lunch break as such. Nannies work through the entire time, getting some downtime but not able to leave to go out for a lunch break.
I have had situations where baby has been asleep and older child needs collecting from nursery/school. MB working from home, so asked them if it is ok to leave baby sleeping (MB takes baby monitor), to which MB may say yes if convenient. In your case, you had a meeting you knew about, so you would not have said Yes to such a request. So did nanny not ask before leaving?
>How do you address this with nanny?
Be very clear with your expectations. Start time is the time they need to be at your home, ready to work. Finish time is when they are likely to be able to leave, and you will do everything you can to make sure they are able to do so. Finish time should be a little later than you know you will usually be available... so if you log off at 5pm... then nanny finish time 5:15pm would be better than 5pm.
Be clear about how to communicate with you during the day. They cannot just be leaving to go out without taking baby with them. They cannot assume you are able to keep an ear out for baby, they have to ask and you have to agree based on your work schedule.
>How do you manage pay?
The contract should have in it about pay for the contracted hours and about overtime.
Avoid clock watching. Give and take works well in my experience. You finish your work early, you let nanny leave early, and pay is the same as if nanny worked the entire contracted hours that day. This builds up some grace for when you are a few minutes late. If you are 15, 30 minutes late, then do overtime pay.
In the same way, if you nanny is consistently at work on time, then on the rare occasion they are few minutes late, you don't mind and don't deduct pay.
You have had a nanny before, so do what worked then. Nip the late arriving for work in the bud and be clear about how you are working from home, which does not mean you can care for baby... nanny cannot go out for lunch.