Let me preface this by saying I'm in neither Basingstoke, nor Reading.
However, I'm a member of a group of expats in the Hampshire, Wiltshire and Dorset area who get together on a regular-ish basis. We do anything from games nights, to lunches/dinners out, to potluck events for the Fourth and Thanksgiving. The vast majority of us are US/UK couples and most of our events tend to be in the Salisbury area as there are a number of us who live here. That said, there are also quite a few who live in/around Basingstoke (who travel to Salisbury for events, mind) and many of us would be more than happy to travel for an event in that area. We just haven't had one scheduled there yet!
I guess what I'm trying to say is if you're interested in meeting up with other people, I'm sure some of us would oblige (myself included)!
Likewise, if you are not opposed to the idea of travelling out this way for a meetup here or there, we always welcome new people.
I am constantly bugging strangers and asking them to join. If you want any more info, just drop me a message. We have a Facebook group, as well.