What I am interested in doing is paying my airfare to GB, paying for my food, but living in an actual historic site for say a week while I volunteer my time as a costumed re-enactor.

As an example I could be living in Barley Hall in York and working there as a woodworker carving a chest or making items.

It would have to be a site that has had some restoration done to the building. I know they are out there...

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That aside there are a few other problems that you would encounter, namely that the people on the receiving end have an idea what level of authenticity you're used to - please don't take this as an insult!

I can'tactually think of any sites off-hand, aside from the one that Regia Anglorum has (just put their name in a search engine and it should come up with their home site which links ncely into it). That's mostly late Dark Ages, up to and including the Norman Invasion - Is this too late for you?
The other main site that has anything like what it is you're looking for is Kentwell Hall, which is a large Living History project in England that recreates what 16th Century life would have been like, unfortunately they did their recruitment for this year already, although there is an outside chance you may get in there.