Totally depends on the nanny.
I now work as a nanny 3 days/week, bringing dd (28mos) with me. Most of the time, I just have the 4 year old (she has two older siblings, 6 and 7.5). I worked for this family before I had DD, as well, so, I know the routine.
We almost always do a big outing on Mondays (when the older two don't get home until 5pm). Museums, zoo, etc.
Tuesdays and Wednesdays are closer to home outings. Library (I have the schedule of pre-school story times), town rec center, local playgrounds, etc. Mom sometimes schedules her for kindermusik classes, depending on the time of year. There are almost always other kids at those places, especially if the weather isn't miserable (I am a person who believes in lots of outdoor time). And when the weather is crap or whatever, the 4 year old and my daughter play together quite nicely (they're 20 mos apart).
Instead of asking leading questions, I'd ask the nanny something along the lines of "How would you plan a day with my kids". Some parents DON'T want the nanny taking the kids out (I don't get it, but,
), but, you ought to get a more honest answer that way, and be able to weed out the "homebodies". Then, I think you need to do a couple things: find some activities for your child - it doesn't have to be formal, but, ideas along the lines of "Oh, J has been going to this playgroup on Fridays at 1030 am, and I'd like for that to continue, and the library has story time for his age group at 1pm on Tuesdays.". Also, you do need to let the nanny know that she has some autonomy - some days, my kids are wound up and I know a trip to the playground is definitely needed. Or Maddie will specifically ask, "Can we got to X today", and, if there's no reason not to, we do. Other days, they may be more low-key, or it's raining or something, so, I grab the museum pass, and we head for the children's museum or the science museum.
Final note: I nannied for a different family with older and less kids when Katie was 3-12mos old - (the logistics of 4 young kids wouldn't work at the time, due to car seat issues). One thing that helped immensely was that I baby-wore. Please make sure you have a comfortable sling/wrap available for your nanny to carry the baby in - it's so much easier to get out and do things that way, and my DD would take her naps in it while I took the girls to the park or museum or whatever, so we didn't have to worry about being back at X time for naps and what not.
One more thing: I was new to the area when I started nannying. One thing that was especially helpful was a list of ideas of places to take the kids, and, more importantly, directions from their house to the locations. That was great. There were some places I probably would not have heard of, and navigating through a city would have been challenging without directions (not to say I didn't occassionally get lost anyway, but, it was good to have the directions).