"I do not understand the pleasure of being so famous," she says. "I just don't get it. I love acting and doing the work. That's what it's all about for me." This is typical of Alexander. Her quiet dedication has led to her becoming one of Britain's top comedy actresses, with a CV that reads like a timeline of British television comedy during the past 10 years. Appearances in Drop the Dead Donkey and Red Dwarf led to a stint presenting The 11 O'Clock Show and eventually to the roles that made her a household face, if not name - as a regular on the female sketch show Smack the Pony and as Susan in the dating comedy Coupling. Lately, she has appeared in the deliciously surreal Green Wing and in The Worst Week of My Life - a British Meet the Parents.
When we meet, Alexander is about to jet off to LA for the second time this year, keen to capitalise on the new American love affair with British comedy. In fact, Alexander believes that she helped to set the ball rolling. "Coupling started this new wave of Americans loving British comedy," she says. "I'm recognised in America from Coupling much more than I am over here. Everybody loves it because it's rude and gets away with it." (While Coupling on BBC America garnered a cult following, the American NBC remake fared much worse. Only four of 10 episodes were aired before it was axed amid accusations of gratuitous smut.)
This year, she made a conscious decision to "break America", "not with any desperate desire to be a huge, famous film star but to check out American comedy". The Simpsons, Seinfeld and Friends are old favourites. "I thought, 'I'd like to go and meet these people'," she says. "So I did. It's very exciting, very exhilarating. If you want to meet up with someone, the chances are you can. You can't do that here. We're not that organised."
As a result of her networking, Alexander landed the part of an idealistic British teacher in the US adaptation of Teachers: her character, she says, is "attempting to be Mary Poppins and getting it wrong". If that sounds like stereotyping, she also road-tests her American accent in the upcoming Amy Heckerling film I Could Never Be Your Woman, playing the part of Michelle Pfeiffer's PA. "I'm a wannabe Michelle Pfeiffer in terms of wanting her job, her life, her man. I'm a high street Pfeiffer."
Alexander's approach to her career has always been direct and uncomplicated. She grew up in London and attended The Godolphin and Latymer School in Hammersmith. "I couldn't wait to leave," she says. "Life has got better and better since the day I left school. I felt like I wasn't comfortable in my own skin." In her lunch breaks, she hung around the stage door of the Lyric, watching the actors go in for matinée performances. At 19 years old, she set off for the Edinburgh festival and hasn't been out of work since. "It was very uncomplicated, really," she says. "I just knew what I wanted to do. As soon as I could do it, I did." She's not sure that her father, a television producer on Panorama who died when she was at school, would have approved. "Had he been around, he probably would have created an almighty stink about it," she says. "I like to think he's looking down and giving me a thumbs-up now."
Her big break came with Armstrong and Miller, a Channel 4 sketch show. Alexander met Miller on the set of an advertisement for Kodak disposable cameras. She played a bride in a "frou-frou" wedding dress, which made her feel "horribly uncomfortable". The two got talking when she stuck her Dr Martens on under her dress for a joke.
Alexander is striking, with her bright blonde hair, cat-like blue eyes and wide mouth, although she claims she is only recognised if she wears "lots of make-up". Surprisingly, the idea of poking fun at her appearance and disguising herself comes up repeatedly. She'd only do stand-up "in character - not as myself, I never want to be myself", and although she has a good soprano voice, she sings only "in a jokey way - it's that thing of not being me".
She explains: "The appeal of comedy is that you're not going to look your best. Dressing-up or dressing down is something I love and feel very comfortable doing. I feel at my least comfortable when I have to look at my best. I suppose I'm lucky, but having to look good is a pressure I don't particularly enjoy. Whereas dressing up, putting on a funny hat or wig is infinitely more amusing."
She indulged her passion for funny hats with zeal on Smack the Pony, which remains her proudest moment. "We were very aware that we were all women," she says, "but we tried not to make a feature of it. We just wanted to be a sketch show starring girls being funny." She believes that nothing has come close to it since. "They're all rip-offs of Smack the Pony," she says. She hopes that the next big thing in television will be a return to mainstream comedy. "It's been quite bleak," she says. "They're still re-running Men Behaving Badly and Only Fools and Horses for God's sake, and that's over 20 years old."
The Worst Week represents her foray into mainstream. While the first series focused on her disastrous nuptials, in the second she gives birth. "It's quite odd, particularly because I haven't done either," she says. "It's like a rehearsal, possibly. I definitely walked away thinking, 'I'm never going to get married'."
Alexander is currently single, her relationship with veteran actor Gerald Harper, 42 years her senior, ended in 2002. She guards her privacy fiercely. "I don't really enjoy the whole celebrity scene. I've poked my nose in a few times, but I don't find it stimulating or entertaining." Her friends, "some more celeb-y than others", are mainly comedians. "All the comedy people in London are pretty much all friends," she says. "People in comedy are just gorgeous, just the best human beings. They are naturally interested in other people and in playing something other than themselves." She even spends her spare time watching comedy ("sadly").
As for the future: "I wait to see what next year will bring. Variety is the key." With that, she heads out into the rainy London street and off to the LA sunshine.
The second series of 'The Worst Week of My Life' begins at 9.30pm on BBC1 on 17 NovemberReuse content