This mum is a CF, she invited her child round for these sleep overs/play dates and organised it all. She clearly just wanted to be able to nosy into @MyLuckyTiger 's lovely big house. I'd be wary of her. I'd absolutely stay home next time there is one of these play dates, give the nanny the night off, then when CF turns up, answer the door, and say "oh collecting Olivia already? I wasn't expecting you until 9pm!" Then call Olivia, tell her her Mum's here to collect, all whilst holding the front door so CF is still stood on the doorstep.
Long term, I'd reduce the frequency of the play dates, and ALWAYS be home for them, always following the keeping the mum on the doorstep whilst you call for Olivia to go now.
I would have a chat with the nanny, just to casually mention you've found out CF has been coming over just after you leave with her toddler and staying at your house etc., joining them on trips out, and you're uncomfortable with this as it was never agreed with the other mum for her to be in your home, and for you to pay for her food etc. Please do not blame the nanny for this, but just say that you don't want the CF mum to do this again, so ask her not to let this happen again. I'd actually apologise to the nanny for being in this position, tell her I'm sorry you've been put in this position, I'd had no idea that this was happening.