Fanny:
In Australia fanny means vag, and in USA, arse.
Chick from USA over here, at the office we ask her how things are and she says “I’m working my fanny off”. We all look a bit peturbed, thinking she is telling us she is a part-time prostitute.
Root:
In Australia root means sex, in USA it means cheering someone on. In Australia you really don’t want to ‘root for your team’ unless your partner and you are both really into sport.
Others?
Pecker:
In the UK, it means “nose” (probably). On this side of the Atlantic, it means “penis”. Imagine my moment of shock when, in Harry Potter And The Order Of The Phoenix, Fred and George say they “kept their peckers up!”
In what scene did they say that?
Merci=thank you.
But in response to a question, in english ‘thank you’ means ‘yes, thank you’ and in french, ‘merci’ means ‘non, merci.’
Randy=American male name
Randy=horny
I’ve always liked that one.
Knock up=British wake someone up
Knock up=American get her pregnant
I’ve known Americans who didn’t know the British meaning who were truly horrified by it. 
I wish I could find the page number, but I am moving on Monday and all my books are packed away for now (sob… I miss them). They may have changed the wording for the American version (if that’s the one you have), because I know they do that.
Since I live in Canadialand, I get the Brit editions, with the slang intact.
I’m pretty sure they are in the Gryffindor common room and Fred and George are trying to reassure Harry about getting his OWLs, so it would be in one of the chapters right before Harry takes his exams.
An old one, I’m sure, but a long time ago I learned this:
Fag (British) – Cigarette.
Fag (USA) – Male homosexual.
When I was in England many years we were in a restaurant talking about someone’s new sports car.
My boyfriend at the time was with my and he was Irish.
One of my my English friends (a girl) invited him outside for a shufty.
I don’t know what he was expecting but he was out of his seat like a shot!!
Spike: Randy Giles?! Why not just call me “Horny” Giles or “Desperate for a Shag” Giles? I knew there was a reason I hated you!
Giles: Randy is family name, undoubtedly…
Covered recently in a MPSIMS thread.
Faggot = American homosexual
Faggot = British meatball or sausage
My favorite:
Rubber = American prophylactic
Rubber = British rubber band (or elastic)
Mildly off topic:
In Denmark there will frequently be a sign above freight elevators reading:
Godselevator
Gods = Goods
Still a bit of fun to point out to the devout.
Getting pissed: British - getting drunk
Getting pissed: U.S. - getting angry
actually “getting pissed” can mean both of those in England. In conversation it is obvious which one but in writing it can get a little confusing!
Pow - A sound you make when you hit someone.
Pau (pronounced the same) - A local word meaning finished.
I love the faces mainlanders give you when you ask them if they’re pau.
There’s also the expression “piss drunk” used in the States. But yeah, not many Americans would interpret “she’s pissed” as “she’s drunk.” Anyway, no real contribution to this thread. When I think of one, I’ll come back. Carry on.
MeanOldLady and Angelic:
I learned from a (oh, the shame) US magazine that Prince William was given shots of alcohol at his 21st birthday party, and stated “you’re trying to get me pissed”, with brackets included by our press (drunk). I don’t know of any popular U.S. terms which could be construed as such different meanings! Sure there’s some out there, though.
During WWII, my uncle was stationed in Britain and got himself embarrassed at a home he was visiting when he mentioned that he had dropped his napkin on the floor.
I doubt that 3 Yanks in 100 have even heard the word serviette (and I do not know whether serviette applies to the ones made of linen), but Yanks put diapers, not napkins, on their infant children. (A more nearly British usage remains in American with the phrase sanitary napkin.)
I thought rubbers were rain gear as in “we leave for shooting at 5, it’s mucky so bring rubbers.”