I'm in the Air Force. We'll get a Cost of Living Allowance over there, and I hope we can save most of that, but I'm not getting my hopes up until I see actual prices for groceries and such.
I wouldn't worry too much. The 5% surcharge everyone pays at the commissary goes towards keeping groceries at stateside costs while overseas. I have always been able to save all my COLA. Overseas is where the real money is made while in the military. I spent 10 years overseas and when I got back, I was amazed at the decrease in my pay.
In England you will be on OHA, so they tell you your housing allowance cap, and they pay your lease up to the exact amount stated on the document. Some people make arrangements for a higher lease, and have the landlord pay utilities under the table.
The roads around the bases are narrow and can be dangerous for biking. I would actually recommend living in one of the more populous cities in the nearby area. Bury St Edmunds is ok, I lived in Cambridge and loved it. Definitely not recommend Thetford at all. I would say the increase in costs to commute is worth the experience of living with the locals in other countries. If you chose to live on base, then it won't really feel any different than anywhere in the US.
The best part of being overseas, is the ability to now stay overseas if you want. You can message me if you have any questions about it.