"Go shut the door" was fully recognisable to me (an Irish-English speaker) and didn't strike me as unusual or sounding particularly American, British, un-American or un-British.
So from experience, I would say it's not peculiar to either.
Of course, 1 is not a statistically valid sample. Since it's a rare enough expression in full though, I decided to look at "Go find out" vs "Go and find out", as it should be more common, especially in non-fiction cases.
A corpus search finds the fuller form ("Go and...") to have been more common in both British and American English, the shorter form to be found in both. In the last couple of decades the shorter form has become more popular in British English but remains in the minority, and so much more popular in American English as to rival the fuller form.
Of course, the corpus only examines written English, and I would suspect that the shorter form's popularity would be greater in spoken English than written. If this was the case then the difference in en-GB* and en-US might be not so much that Americans use it more than Britons, than that Britons are more likely to consider it too informal for at least some written use - it's impossible to say.
*Incidentally, what is with google using "eng_us" and "eng_gb", as if we didn't have an agreed-upon system favouring ISO 639-1 over ISO 639-2 on the web for the last 17 years?