Sound like a whinger? Show maturity? WTAF?!
It doesn't matter what job the boss does. The nanny has contracted hours, and the boss is just taking the piss - especially if she's admitted she's been in the pub! If it had been set up at the beginning, that the boss had unreliable/unpredictable hours, then that's a different matter. But it wasn't like that.
OP you have to deal with this and stop it happening, because this is all about her not having any respect for you.
Don't make it emotional. But send her an email something along the lines of:
"Dear X
Having been employed by you now for [x] months, I would appreciate it if we could review how things are working.
Whilst everything seems to be going well between me and [child/children], I'm concerned about the hours that I am being asked to work. As we agreed, I would start at [x] am and finish at 6pm. However, it is increasingly the case that I am not finishing until 6.15pm. In the last month, for example, I have finished on time on [x] out of [y] occasions.
I am happy to be flexible, but regularly working late without being paid for this extra time is not acceptable. I am wondering if it might be appropriate for us to re-negotiate our contract, and extend my working day (and therefore increase my pay) so that your working day can be accommodated?"
And start keeping a log from now on in terms of your arrival time at work and your boss's arrival time home. Get her to sign them as well, so there's no arguing about it.