YAstillBVU your nanny is doing nothing wrong and you seem like you're nitpicking and complaining over the most trivial things.
she cooked DS (2) a 9 inch pizza for his lunch because she says that's what he asked for. When I asked how much she ate, she said that he ate well over half of it and that she shared that with him. Choice of lunch aside, am I meant to cater for the nanny too?! That to me sounds more like an aupair.
YABU - it's fairly standard for a nanny to be provided with food when they're on shift. I had an aupair when DD was small (worked better for us as I only needed term time help and we lived in a popular "study abroad" location so plenty of students wanting the job to fit round their studies) but when looking at nannies it was perfectly standard to either give the nanny a meal allowance as part of the weekly expenses kitty (so if they took DC to a cafe for lunch after an activity session across town then they could pay for their own food out of those funds too or they could use the kitty money to take DC to the shops to buy bread and ham to make them both sandwiches for lunch if there was none in the house) or tell them they could make themselves something for lunch at home from whatever was in the cupboards as long as it was reasonable and they told you if they used the last of something so you knew to replace it in the weekly shop.
Some nannies might prefer to bring their own food but considering that this isn't like a normal job where you can just leave for a 30 minute lunch break it's a reasonable expectation that if they're expected to make lunch for the child(ren) then they can make enough for themselves too.
Sharing a pizza with DS would be totally reasonable, making him a ham sandwich helping herself to a three course steak dinner would not.
Additionally would you prefer that she had thrown the rest of the pizza away rather than eating it? What difference does it make to you if she ate the pizza rather than it going in the bin? You're sounding very mean and nitpicky to not want to allow your nanny a couple of slices of pizza for lunch.
The advertised role was for a nanny housekeeper. When the kids slept, she asked if I minded her vaping. During nap time, she folded away some clothes for me but didn't empty the dishwasher. The play room was definitely not as tidy when she left as when she arrived. I feel as though my life isn't really being made any easier, I still spent an hour after she'd gone emptying my dishwasher, clearing up the toy room. She left her coffee cup on the window sill of toy room. Small things but not things that make my life any easier!
What's the relevance of her asking about vaping? If it was because she wanted to do it in the house then that would be unreasonable (I don't smoke or vape and wouldnt want the smells in my house) but I assume she asked as she would do it outside so wanted to be sure that you were ok with her having DC napping inside while she was out in the garden. Some parents would want the nanny to be in the house at all times while the child was asleep, even if that was the nanny's only break time, meaning they wouldn't be able to go for a smoke/vape break without the child so it may have been a deal breaker for them with taking the job.
As for the jobs done during nap time, you have a 2yo, so they aren't napping for hours and hours at a time, what exactly are you expecting her to get done in the time your child naps? By the time she's had a break, folded the laundry and tidied the playroom (might not be to the level it was when she arrived but it is being used so you have to be realistic about how much stuff is still left out) what else do you expect her to do?
Considering it apparently took you an hour to tidy up and unload the dishwasher, when do you want her to fit that into the day, considering you also want her out of the house for most of the day too?
Have you given her a list of the housekeeping tasks you expect her to do? If I was doing a nanny housekeeper role without being given a lost of the housekeeping tasks I was expected to do every day then I would assume the employer wanted a nanny who would prioritise the child but do a few household tasks - dishwasher, laundry, tidying etc- when my schedule allowed (ie when the child was asleep). I would be very pissed off to find out you were moaning about me not doing XYZ jobs without explicitly telling me that you wanted XYZ jobs completing.
The coffee cup on the window sill is annoying but it entirely depends on the circumstances - as a one off I wouldn't mind, as it could be that she was tidying the playroom and drinking her coffee while DC was napping and then they woke up so she put the cup down on the window sill to go and fetch DC and then forgot to move it to the kitchen; if it was something that happened on a daily basis then a simple "Nanny, can you please make sure your coffee cups are put in the dishwasher before you go as I keep finding them on the windowsill in the playroom after you've finished for the day, thanks" should sort the issue.