WI: What would the relationship between America and Britain be in the world of Doctor Who?

In Doctor Who, they mostly set in Britain, in episode 12, Series 3, U.S. President, Arthur Coleman Winters basically invited himself in to greet the "Alien Species" the Toclafane in a summit hosted by British Prime Minister, Harold Saxon (Timelord, The Master in disguise as human), Saxon then suddenly took control of the summit and had the President killed on National Television, He had then took over the world using the Toclafanes, too short the story and spoiled the ending, the Doctor managed to reverse all that to only the moment the President gets assassinated, And I am wondering, What would the relationship between Britain and America would be after 2008?

By the way, Let's assume that Arthur Coleman Winters won the 2004 Election against Bush as a Democrat.
They knew it wasn't the real PM so probably much the same as in real life, at least it seems to be on the show. The American government knew about Torchwood and UNIT & I think they cooperated on alien threats.
The problem you have is Barack Obama turns up in the "End of Time" as President.
Yeah, Some say he was Winters' Vice President, I highly doubt that since he was not that well known, I personally believe that there was a Vice President under Winters that becomes President for 10 months before Obama took office