I'm going to belatedly throw in my two cents.
As a hardcore Trekkie, I had heard of Doctor Who maybe... twice, in my whole life, before Enterprise was canceled. When I asked my dad about it at some point, he told me about watching the Tom Baker reruns on PBS in the 1980s. He couldn't really describe it; just said it was a very odd show "with a lot of scarves" and that was that.
I started watching SciFi Friday after ENT went down the drain, though, and thought Atlantis was going to be my "holdover" series until the next Trek series came out. Then one night, I caught "The End of the World" on the SciFi network's second or third runthrough of the first series. I was totally befuddled but reasonably entertained. Totally empty-headed, of course, compared to my ever-so-cerebral Star Trek, and, "Ha! I'll bet those Brits are pretty proud of those cheesy production values," but I tuned in from time-to-time. I had a friend at the time who was pretty into sci-fi, and we got together most Fridays to watch whatever happened to be on. He had broader taste than I did, having seen Firefly, B5, all the Stargates, Crusade, the original Galactica, the atrocious Galactica 1980 (!), Andromeda... basically every American space-SF drama. But he, too, was a total DW newbie.
I think the moment it changed for me, when I stopped being a mildly entertained but mildly derisive viewer was when I happened to catch "Parting of the Ways" at a hotel. Nobody else in American SF was doing this kind of thing -- this huge, zany, but absolutely human adventure. It was the anti-Galactica, and oh how I longed for a good alternative to that depressive's dream. I went through what I imagine British children do in their early youths, all the distrust and hatred for the Tenth Doctor because darn it I loved Nine and his ridiculous ears, and I wanted NINE and Rose together, not this ridiculous interloper!, until of course I finally warmed up to Ten just in time for "Doomsday". (I loathed "The Idiot's Lantern," as did everyone else I knew, and between that and "Fear Her" and then "The Impossible Planet," I nearly gave up on the show completely. What surprised me was that a friend tells me that "Idiot's Lantern" was actually quite popular in Britain. Weird!)
By the time Series 4 finally started, I was well into university, and I had got a couple of friends hooked on DW, so I finally had a good clique to enjoy it with. But I'd encountered other, independent, real-world DW fans nowhere outside sci-fi conventions, and even there they were a great novelty. The vast majority scratched their heads when you mentioned it.
And then something changed. From the few replies I've read in this thread, it looks like that change is a long way from culture-wide, but it was a palpable shift that took place between approximately July 2009 and January 2010, right around the end of the specials. First, I started hearing about other fans: I'd mention this "bizarre British cult show" I watched, and one of my friends from the local Catholic seminary would say, "Oh, Doctor Who? My sister's a huge fan of that?" Or, "I think that's my roommate's favorite show. Confuses the hell out of me, though."
Then I started meeting them. Old friends of mine, who'd always ignored my raves about the show, suddenly had other friends who wanted them to try out Doctor Who, and they converted overnight, with vast, series-long catchup runs spanning all four series, trying to get done before "The End of Time, Part II" or, failing that, "The Eleventh Hour." I'd be sitting in a college philosophy classroom, talking about the metaphysics of time travel, and I'd confidently argue that it's just a "timey-wimey ball of stuff", certain that the reference would go straight over their heads, and some totally unexpected English major -- usually female -- would fall out of her chair laughing. I started knowing at least as many Doctor Who fans as 30 Rock fans. But I knew this was still just me -- I'm a huge SF nerd, and we tend to draw our own kind to ourselves.
But this year was really weird. Suddenly, my youngest sister, the least sci-fi person in the house, who only started showing up for Star Trek viewing parties because everyone else was going, had a school friend who started talking about River Song or something, and recognized it from all my ranting about the show. And she became a rabid fan -- she watched everything in four weeks flat, just in time for "The Impossible Astronaut." I was down in Chicago for a weekend and found the whole El line plastered with DW advertising. I opened the comics section to find the Most Pedestrian Of Comics, Sally Forth, which has never to my knowledge shown the least interest in nerdy topics (unlike, say, Candorville), running a joke whose punchline was about River Song! My sister and her friends went on a band trip and plugged "Rose" into the bus-wide DVD player, and got a bunch of freshmen high schoolers into the show. A friend bought me a Sonic Screwdriver for Christmas -- three years before, I'd ordered one online and, when I put together my Ten costume for Halloween that year, not a single person I knew recognized what it was. I ended up being invited to a Series 6 DW launch party whose entire population consisted of high school girls -- and I had to pass up invitations to two other parties. It's been weird how this show has entered the popular consciousness of my age group.
It's now got to the point where I can assume that in any given college classroom there are at least two other Whovians in there with me. Some of them still haven't seen Series 5, because it's not on Netflix Streaming yet and BBC America is premium cable (so even among the upper middle class, most people don't have access to it), but they gush about how they cried when Ten died. (For most of them, Ten was their first Doctor, and it's been fun watching them rage against Eleven for the first half-season, just as I did when Nine turned into Ten.)
It still feels like we're speaking a secret code. Most Americans -- even college kids -- still haven't heard of Doctor Who, and are totally baffled when you start talking about TARDISes. We're as far away as you can get from the immense cultural penetration DW has in the U.K. (where its audience shares regularly exceed what American Idol gets Stateside, which is always pretty incredible for me to think about). But the series is growing in popularity, quite quickly, and it's been thrilling to watch it happen. I imagine this is what it must have felt like to be a Star Trek fan in 1970, right after cancellation, just as the great sweep of syndication began. Except, unlike back then, I still get to enjoy new episodes right along with everybody else.
Two things to remember: (1) the 1.3 million viewers on BBCA wildly underestimate the total audience, which as far as I can tell is mostly via Netflix, and (2) as far as I have experienced, there are only two types of Americans: those who've don't know what Doctor Who is, and those who can't get enough of it.
EDIT: I should add: nobody has a clue about the old series. In America, the way to mark yourself off as not just a Doctor Who fan (which is cool and kind of trendy, to those in the know), but as a Huge Honking Nerd (which is less cool) is to start talking about something like the Hand of Omega or Genesis of the Daleks. I'm the Classic Who Intellectual Heavyweight in my group, because I've watched "Battlefield," the TV movie, and the first parts (but not the full serials) of "Mawdryn Undead," "The Three Doctors," and "An Unearthly Child." Oh, and I read that Ten Doctors webcomic which caught me up on all the old companions, which was hella useful when Jo showed up in SJA and everyone was, like, "Who's this Jo woman?"
On the other hand, they all asked the same question when Sarah Jane showed up in "School Reunion." I sure did. We'd never heard of her! I imagine that would sound strange, to a Brit. They figured out pretty quick what was going on, though, and loved her. Some of them were quite surprised to hear that she was actually a character in the old series, not just someone RTD made up for the occasion! (And they were even more amazed that K9 had once been a regular!)
EDIT 2: Oh, God, I didn't realize this was from last August. I thought it was from last week! I apologize for the necropost.