Back when I first fell headlong into Doctor Who, in 2013, I was working for a different development studio than the one I am now (actually, it was three companies ago over the last eight years - if you want job stability, independent software development studios are not a good choice) and one of the developers, Ian, was English. Ian was a little older than me (read: old enough to have grown up watching the classic show since the Third Doctor, possibly the tail end of the Second) and sported a thick accent which most of us Americans often couldn't decipher, even ones more comfortable with non-RP accents than I. I have since learned to recognize his accent as something perhaps Midlands, similar to Matt Smith.
When I started talking about DW at work, Ian took it upon himself to educate me to be a proper fan. However, only one thing he told me stuck with me, and it was this: "Never refer to the Doctor by number - the Third Doctor, the Seventh Doctor, the Tenth Doctor. We use the actor name - the Pertwee Doctor, the McCoy Doctor, the Tennant Doctor." He implied (or maybe he stated it outright; it was eight years ago, after all) that using numbers would mark me as an American (as if that wasn't obvious already) and an ignorant wannabe fan.
He does have a bit of a point, since after all, we refer to the Delgado Master, the Ainsley Master, etc., rather than the First Master and the Second Master (though we also refer to Romana I and Romana II). However, in eight years of hanging out in this corner of the fandom, I've never heard anyone, British or American, consistently refer to the Doctors by the actor name. Occasionally, yes, but not consistently.
So is/was that really a thing? There are a number of possible explanations. First, most people I'm in contact with in-person and on the web are American, so I mostly see American phrasing; however, I do have a number of British friends, many of whom are reading this right now, and they tend to use number rather than name. Second, since I'm mostly active in the fanfic community, I see the phrasing that's used there, and it's number; you won't see "Eccleston Doctor" as a selectable tag on ffnet, Teaspoon, or AO3. (Ship names would also get pretty complicated, if you can't use "Ten/Donna" or similar.) Third, Ian's been living in the States for a very long time (he's married to an American and has been here all that time), and perhaps the convention in Britain has changed and he's just not aware of it. Fourth, maybe he was just winding me up.
Whatever. It's not like I'm going to change now. :)