How many classes/activities a week would be normal?
We do something every morning. Playgroups a couple of times a week (about £2 a time), a singing group (£3-5) and baby gym (£5ish a session, book for the term). The older ones also do afterschool activities once or twice a week (currently ballet about £5 a session, book for the term). There is a huge choice of things to do (we're in London) and a lot of them are free including a playcentre, free playgroup, library singing/storytime etc. so you could do a couple of things a day and not spend a penny! Depends on what you ask the nanny to do (though he/she may begrudge not being able to spend anything!)
I would probably consider leaving him in nursery (private or state) for 3 or 4 sessions a week anyway, I like the aspect of socialising that nursery gives.
You should be getting 15 hours a week free nursery education, so you could use that to keep him in pre-school/nursery 3 sessions a week. I wouldn't worry about the socialisation too much though, any good nanny will organise playdates and take outings to playgroups etc. where there will be other kids his age to play with.
a nanny would be logistically sensible. Is that a good assumption?
Yes. When you have two children, needing to be different places, at different times a nanny or childminder would be easier than a nursery as they will pick-up/drop off as required. Another thing to think about is that if he's at a nursery he won't be able to have his new school friends round to play after school, or join in with clubs that go on afterschool, whereas a nanny will cater for this.
What other 'hidden costs' do I need to think about?
The only thing I can think of that hasn't been mentioned is 'wear and tear' on your house - if they're doing playdough/paint/cooking/sticking etc at home instead of nursery you may find your cleaning bill increases!
Our day looks like this (actually it's changed very recently, but this is how it looked for the previous school year).
8am - finish up breakfast, dress the youngest, get everything ready for school.
8:30am - leave for school (drop DC1 and DC2)
9-11am - activity with DC3 (gym, playgroup, library, singing etc)
11:15am - 2:30pm - pick up DC2 from nursery, go home,prepare lunch while children play, have lunch, DC2 has 'quiet time' and DC3 sleeps. Tidy up, put washing on, sit down!
2:50pm - leave to pick up DC1 from school
3:20pm - 5:00pm - pick up DC1, go to the park/library/after school activity/playdate or any errands (e.g. haircuts, new shoes, dentist, doctor etc) or do an activity at home (e.g.craft/cooking)
5:00pm-6:45pm - prepare supper while children play, eat, tidy-up after supper, bath, pyjamas, warm milk and story (don't always get to the milk/story bit!). Parent(s) come home and I finish.
Costs tend to be slightly higher in the school holidays as we go out on 'special trips' softplay/aquarium/IMAX cinema/funfair/zoo etc
I've worked in a nursery and am now a nanny and in imo if you can possibly afford a nanny you should have one. I know this isn't a available as a choice for eveyone, but if it's comparable in cost, then I really feel the overall quality of care is higher with a (good!) nanny than it would be in a nursery.