Hi there, are you in North Finchley? You could try Toddler Sense, or Boogie Babies, or Tumble Tots (google them in Finchley and you'll find them) or the local libraries all have various stay and play activities. Have you looked on the Things To Do/Baby and Toddler Groups section on Mumsnet Local? Plus you could try the artsdepot cafe NCT coffee morning on a Thursday at 11. It's just an informal gettogether for mums, it doesn't matter if you have a teeny baby or a toddler, Lorraine who hosts it is really nice and will introduce you to other people. It has a little soft play area and loads of space for buggies too. Hope that's helpful - if you do a good one, you could always add it to Mumsnet Local too and put a review on for the next new arrival.
North London must be a bit of a change from Seattle! There are a few things on at North Finchley library for toddlers. They do a stay and play session on Tuesday mornings and afternoons, Story Explorers on a Thursday afternoon, and read and rhyme on a Friday morning.