I can't speak with any authority on which surnames are the most common in Great Britain. Alls I know is that the St. John-Mollusc family tree has died out, thanks mostly to the annual Upper Class Twit Of The Year awards.
Even so, I don't think a surname like "Fanshawe" could be that commonplace in the UK. It recently popped up on American TV screens last week as the name for the British Ambassador and his son. (By the way, as sainted a memory as his late wife might enjoy, Ambassador Fanshawe really should get a paternity test on that d-bag kid....)
So it's pozz'ble, just pozz'ble, that Ambassador Fanshawe could be descended from Harry Fanshawe, from episode one of the historical mini-series 'The Devil's Whore'.
Harry was soon unjustly killed off in that opening episode on orders of Charles I's government and the rest of the series dealt with his widow Angelica and her taking up the cause of revolution, bringing her in contact with historical figures like Thomas Rainsborough and Oliver Cromwell.
I haven't seen the mini-series; I hadn't heard of it until I started research into the name "Fanshawe". But I don't think Harry and Angelica were married long enough to have children and I don't think he left her with child before she died. But there's always a chance that, despite his love for Angelica, Harry did have his way with some saucy tavern wench in the district and saddled her with a bastard. And this unofficial Fanshawe could have begun his own family tree which would eventually lead to Ambassador Richard Fanshawe*.
* I'm seeing the name spelled as "Fanshaw" at certain sites, notably the IMDb, which as we all know is always correct. However, in the episode we saw the layout for the ambassador's residence labeled as "Fanshawe Mansion".