I’m watching the Daleks take Manhattan (or whatever it’s really called) episode of Doctor Who, and [del]my girlfriend[/del] Martha just referred to a “service elevator”.
I am agog. Shouldn’t it be a service lift?
Maybe the Tardis’s Universal Translator made her say " elevator."!
Perhaps they have a version aimed at [del]barbarian foreigners[/del] Americans and others who don’t understand the Queen’s English.
We’re bilingual - we speak English and English, thanks to all the American TV programmes/programs we’ve seen over the years.
UK TV producers do have one eye on the lucrative US market, and have done for a long time. When the US networks started buying 60s show The Avengers they insisted it was in colour (previous series/seasons were in black and white), and the title sequence trumpeted “In Color”. Sic.
Both words are (probably/near enough) used just as much as each other in the UK. ‘Lift’ is a far more informal/colloquial/relaxed way of referring to it, whilst ‘elevator’ sounds a little more technical. Also, I can’t imagine the Queen saying ‘lift’ at all…
Only where it’s a lift that’s for goods rather than people. When it’s for people, lift is much, much more common - elevator would really stand out. So the phrase ‘service elevator’ sounds normal to me, whereas I’d be surprised to hear an English person talk about getting an elevator to the tenth floor.
15 years in the UK, and not once have I heard a Brit call a lift an elevator. YMMV.
Actually, a surprising number of them seem to think the American word for lift is escalator. 
Sort of like the while/whilst dichotomy? 
Yes, this is it in my experience. “Elevator” only tends to be used in combinations like “service elevator”. “Service lift” sounds a bit odd, but not as odd as calling a lift for people an “elevator” would.
I am British, but when in America, or speaking to Americans, I try to remember to speak American rather than English. It seems like the polite thing to do.
She knew she was in America, so it’s a frikkin’ “elevator”! :D:D:D:D:D