A female Doctor.

[Nekojin]

I don't see why you couldn't have a non-British actor play him. Even if you insist that The Doctor have a British accent, that's no real roadblock - it goes the other way often enough. Between Bob Hoskins and Hugh Laurie playing Hollywood-accented Americans (and many, many others), there's no reason to think that an American (or someone from damn near anywhere around the world) who has had a proper voice coach couldn't do The Doctor.

[Ginger Tea]

TBH I don't think the BBC could afford an American actor, least not a well known one.
Classic who suffered budget cuts all the time and although new Who is popular and distributed world wide, it is still funded via the licence fee.

Jonathan Ross was paid quite hansomly by British standards to remain with the BBC a few years ago and although popular and drawing in the ratings, he earnt more than some shows were budgeted.

The cast of Friends were allegedly on close to a Million Dollars each per episode towards the end, I don't think we've had a show that's had a budget that high set aside for it EVER.

ITV is different, it gets money via adverts and sponsorships, one of the reasons the BBC lost out on too many sports was SKY and ITV could out bid them, to secure the rights to one day international cricket, it could cost the budget of x episodes of a show.
I am not saying the other networks have cash to burn, but they have other ways of gathering income, not just selling rights abroad and DVD box set's which the beeb can do too.

[HYHYBT]

And it happened to such an extent that you now have people refusing to watch the show on the "not another white male" principle.

Yeah, like that's going to last five minutes once there are new episodes to see.