Would BBC Hire an American to be the Doctor?

Probably not. Doctor Who is a production of the BBC, which is completely government owned. Their charter would most likely require them to hire British actors in the major lead roles. This would most certainly be the case with Doctor Who. It is the only production the BBC makes completely in house.

AMC's Walking Dead has several British actors playing Southerners rather well, so it isn't just Hugh Laurie.

But I doubt the BBC would consider an American actor for The Doctor. The British are a proud people and Doctor Who is quintessentially British, thus requiring a British (or Scottish) actor in the role. Even Capt Jack Harkness, although an American character, was played by Scottish-born John Barrowman.

no . Hugh Laurie would be way out of their price range