Oh, PP from ^^ - definitely get a log notebook. You can use it to leave notes that you might otherwise forget to tell your nanny at drop off/pick up time and it's a good way to see the schedule of naps and feedings.

Your nanny should have already said that this is a useful item to have.

I don't think your kid needs to be one on one, but because of our experience with a bad nanny (wonder if it's the same one!), we hired our own nanny to work 1-v-1 with our son and then looked for a share. It's worked amazingly and we're thrilled and love our nanny. So it's very possible!

OP - we had a nanny like this for our first nanny. Honestly, trust your gut.
The nanny wasn't actively trying to feed our baby ("he must just not be hungry"), she was leaving him in his crib for hours (he would wake up and just hang out, not crying, so she never checked on him!), when we started a share, she let the other child sleep in the stroller after rocking him back and forth in it to sleep even after the parents told her multiple times to NOT do that. She was taking them out for hours at a time in polar vortex weather and wasn't changing their diapers often so my kid got diaper rash. Our child was miserable and losing weight.

She lasted 5 weeks - we gave her the benefit of the doubt that there was some adjustment period. She was just a terrible nanny and I wish I'd trusted my gut sooner.

So I say this to trust your gut. The nanny is hired to do what you outline (with a little room for change once the nanny is there a while). If she's not doing it after another week or so, don't stick with it.

One thing that has worked very well for us in a share is babies of different ages. I was skeptical at first, but a 6 month age difference allows one child to be a little needier and the other to be more independent.

Bottom line, give it a week and trust your gut.

I think your baby needs the one-on-one attention from a single nanny at this point.

A nanny share just won't work because it is really theoretically impossible for just a single person to effectively give that much attention to your child AND care for another child as well.

Curious what do you know are your baby's sleeping and eating cues?

Posted before finished.

For the eating
1. Is your baby using a bottle/ nipple he is familiar with?
2. Have the bottle warmed to temp so it matches the temp of your breast milk.
3.Provide a shirt or blanket with your scent to use when nanny is feeding
4. This has always worked really well with eaters my non eaters have her position the baby like you would for when you are nursing him

As he becomes familiar with her 3 and 4 will phase themselves out.

5. If you can on weekends have someone else give him at least one bottle.

I suspect the lack of sleep is a function of not eating enough he cane sleep because he is hungry and he is turn too tired to eat.

If she can sort the feeding out the sleep will follow.