Hi. Am struggling to think off small baby related stuff on a Tuesday afternoon but I am sure that someone wise will be along with suggestions ( I work on Tuesdays so never do baby stuff then). The York community gymnastics foundation is fab for when they are two - it has a waiting list and is in Heworth. There are various groups around like Songbox, Toddlersense and Munchkin Music - maybe you could lookup their times on the web or in the Library in town. I know there has been mention of a York meetup but that seems to have gone quiet at the moment - but you could keep your eyes open for that.
It can be quite a struggle to feel settled in a new place and I found getting out and meeting people helped me when we moved up - like you we had no local friends or family when we moved. hth.
Hi Toobreathless, I have a ten month old boy and would be happy to meet up with you. We've found some good things to do in York, it's a lovely place to live. I don't know how I can direct message you but if you want to email me we can arrange something? Angela at solltd dot com