Hi
I don't often look at this part of the site, so just saw your message and realise it was nearly a month ago...so apologies for the late response. You may even have moved already.
We live in Gosport (Alverstoke) not far from Haslar marina, so hopefully I can be of help. We also lived in London until 4 years ago, so we made the same move. It is a wonderful place to live with children and we are really happy living here...so I think you have chosen well :)
Our dd is nearly 4 and ds just 2, so similar ages. They go to Alverbridge nursery and pre-school, and have been there since our dd was 14 months' old. We think the staff are great and both our children are very happy there. They do get full though, so it'd be best to contact them directly if you can (not sure how to do links but google Alverbridge nursery.) They both go for 2 full days per week (I work 2 days). On the other 3 days I look after them, so I don't know much about nannies. However, I do know some childminders (one who was previously employed by the nursery and so formed a close bond with our dd.) I'd be happy to give you her name if you pm me.
There are other nurseries and pre-schools locally, but I don't have much experience of the.
There are a lot of groups within walking/cycling distance. Not sure if you cycle but it is flat here, so can be a good way to get about. You can also take bikes on the Gosport ferry (so it can be a good way to get to Portsmouth/Gunwharf Quays.)
One toddler group run by the 'NCT Gosport and Lee on Solent' (there is a facebook group of the same name, which if you join you can see all the activities) meets every Friday morning at different locations for craft/nature walks etc. It is called Little Explorers and is a great group.
There are also different classes for preschoolers, some of which may be interesting for you.
In the Alverstoke Tennis club there is a preschoolers 'Set4Sport' class at 9:15 on Sunday mornings. It is a 'turn up and pay' type class, which is good if you don't want to commit to a whole term's fees up front.
Swimming classes locally are run by Aqua Academy at various school pools and on various days.
The Gosport Discovery Centre (library) is really good and has a cafe inside, so can be a great place to take toddlers. They do rhyme time and story sessions, as well as other events (Easter crafts etc). Again, you can join the facebook group to be notified of things.
There are several children's centres and I think some stay and play sessions are organised which take place on Stoke Road in the Methodist Church, but I don't ever seem to get to these.
St Mary's Church (Alverstoke) run a monthly 'Messy Church' which is usually on a Thursday afternoon and includes Christian themed craft/singing etc. It usually takes place in the infant school on Ashburton Road.
St Francis Church Hall would be quite near to you and I think this has a weekly toddler group, but I'm not sure of date and time.
Hope this is all useful and good luck with the move. Obviously, as well as all the organised toddler groups there is the beach, crazy golf, outdoor paddling pool at Stokes' Bay seafront etc for kids.