Nanny here. Yes you could argue I am biased.
1) There is no developmental reason for a 2yo to be in school. Whatever they get out of it at 2 they will get out of it at 3 and probably better and with less stress.
2) I would not add a big adjustment right after adding a new baby either and finally,
3) the first time nanny/SAHP kids go to school they end up catching EVERYTHING. I would avoid that with a newborn, and not because of COVID. Small kids share germs back and forth easily and a cold or fever for a toddler could be really serious for a newborn.
As for the nap thing and worrying about getting the baby on a schedule:
What I have always done is to try to get the younger sibling on a 3-nap-a-day schedule by about 4-6 months. Get everybody up and dressed and fed breakfast, then head to a park, playground, play in the yard, go for a walk, etc. Once we get there I pop the baby in a carrier and keep a bottle handy so I can feed the baby as needed and then let baby pass out for am nap in the carrier. We play outside until baby wakes up, then head home.
At home we do indoor play for toddler/tummy time for baby. Then do snack/bottle and I set up toddler with a device or other distraction for 20 minutes so they can sit on the floor in baby’s room while I settle baby for a proper afternoon nap in a crib. Then get toddler down for their nap.
Baby usually wakes before toddler so they get some extra snuggles and one-on-one, then when toddler wakes we do an indoor activity like an art project together, with baby in my lap or on a play mat nearby. Usually around dinnertime for toddler baby is starting to get fussy so I will wear the baby in a carrier for a catnap during dinner, then wake baby up to take a bath when I take toddler upstairs for a bath. After everyone’s bathed and in pjs, I will snuggle with a pile of books and feed baby while I read books to toddler. Then baby is ready to be put down and subsequently do a final bedtime for toddler.