The 1991 John Goodman comedy King Ralph followed a boorish Las Vegas lounge singer whose grandmother had an affair with a British prince. When the entire royal family is wiped out in a freak accident, Ralph is the last living claimant to the throne. Throughout the course of the movie, Ralph befriends a newly crowned African king who wants to create the first African automobile company, fails to negotiate resource rights with the Kingdom of Finland (IOTL a republic), uncovers and suppresses a Stuart restorationist plot, and gives a rousing speech to Parliament before abdicating in favor of another long-lost claimant. Ralph ends the movie as the Duke of Earl, heir-apparent to King Cedric I, and father to the future King Ralph II.
A few things off the top of my head.
1: An oafish American holding the British throne even for such a short time would discredit the monarchy in the eyes of the British people. When Cedric's reign comes to an end in 2013 (based on when Peter O'Toole died IOTL), his potential heirs will be Ralph I, the American who had previously abdicated, and Ralph II, the 22 year old son of a former Vegas lounge singer and his working-class consort. It's possible that there would be a stronger republican movement in the King Ralph universe. Because Ralph's infant son is in the credits as Ralph II, its implied that this movement fails and Ralph II does indeed become king. Instead, I think that both Ralphes' worldliness and humility would further endear them among commoners despite driving a wedge between the Crown and the aristocracy. Scotland and Northern Ireland might not take so kindly, but that's hard to judge because they're completely absent from the movie.
2: As king, Ralph I took a fairly active role in politics. He unilaterally negotiated investment in Zambezi (the fictional African kingdom mentioned above) in a deal that provided factory jobs in England. His son likely grows up in the lap of luxury, but with the working-class sensibilities of his parents. As such, Ralph II will likely take a more active role in politics, particularly to improve the status of blue-collar citizens.
3: The investment in Zambezi will legitimize King Mulambon's reign, implied to be unstable due to how recently he took the throne. Zambezi will succeed in creating an Africa-based automobile company and will be considered one of the great exceptions to the political and economic instabilities faced by many post-colonial states. Anglo-Zambezi relations will likely remain strong. Between the Ralphes' American heritage, spurning of the aristocracy in Britain and abroad (see Finland above), and ties to African states, the United Kingdom will further focus its foreign policy outside of Europe. We'll see an earlier Brexit, probably with a much wider majority voting to leave.