Hi, I´m from Burford originally, but haven´t lived there since having kids, so not much use! Witney has a great indoor play place, in Waitrose car park, that might be a good starting point.
Big, fab shopping centre at Swindon with massive train museum thing next to it, lovely for little boys. There´s nice big parks in Witney too.
Somewhere near Stow is an old steam railway, which has engines running most of the time, will ask parents what its called and let you know.
Great pick your own place called Millets, think it´s near Abingdon somewhere, has lovely cafe, lots of animals to look at and big farm shop. I´m sure it has a website, but will find you more details if you get stuck.
Cotswold Farm Park somewhere in Gloucestershire.
The Fox at Barringtons, lovely pub right by a river (no fence though) which is child friendly (slight contradiction....!), ie they let kids in, have a footy table and big open area for running around, don´t mind you throwing stones in the water, do kids meals etc etc.
There are definitely baby groups though in and around the area - have you tried the library, they usually have some sort of things going on....
Would love to meet up, but we live in Portugal now, so could be a bit tricky..... Back at Xmas time though, so we could meet up then!
HTH, will keep posting on here with ideas. Lucky you though, its a lovely area once you find your feet!
x