Hampstead is lovely and absolutely heaving with young families. I don't know of any specific playgroups, but most of the libraries will run free Rhyme Times on a weekly basis. There is a Camden library right near Keat's House. Try checking out what day they do a Rhyme Time. I know there is a popular music class a bit further afield in Hampstead Garden Suburb. It might be called Monkey Music. It may be a bit of a walk for you depending on where in Hampstead you have your flat, but it's such a lovely area that it will be a nice walk.
As far as the restuarants go...Try the Creperie when the line isn't too long. It's a "permanent" street cart that sells the most fantastic crepes. There is almost always a line. Cafe Rouge, Carluccios and Giraffe are great ones for kids. They are very tolerant to messes etc. The Old Bull and Bush is a great pub for a night out. Fantastic food and good crowd later in the evening. You don't get many young kids or rough types. I haven't been on a weekend for lunch, but if it's anything like it's sister pub the Orange Tree, it will be family friendly.
Definitely check out Golders Hill Park. It's part of Hampstead Heath, down West Heath Ave (rd?). It has a small animal enclosure and a good playground.
Except for all the hills you will have to walk with the stroller, you will love Hampstead.
