I say it very much depends what Britons you are talking about. It is a dialectical style, found in a few places, but the most famous and that which tends to travel internationally is the East London, Cockney Rhyming Slang.
Now, most Londoners know a fair bit of it, growing up hearing it. But it depends what area of London you were brought up in, and more importantly what class you were. A lot of my friends use quite a bit of it, but not continuously like you see on films, it crops up as slang words, in the way that any other colloquialisms do.
Often, people do not know the etymology of the words they are using and do not know the rhyming part. Commonly used words such as barnet, boracic, china, cobblers, mickey, scarper and butchers are used frequently, but few know the rhyming component for them, they just use the words. Cobblers and mickey are more widely used than just London, not sure about the others.
As requested by Robusto, a brief set of rhyming expansions for the words I quoted above.
barnet -> barnet fair -> hair
Barnet Fair, was a regular horse trading fair held in the town of Barnet (my home town!), and well known by Londoners, as it was on the Great North Road, and was the major horse buying location for the capital. Further back it was the largest cattle fair in the country too.(Wikipedia)
boracic -> boracic lint -> skint
Boracic lint was an often used dressing for wounds, lint soaked in boracic acid. It rhymes with skint anyhow, which means pennyless, broke. Note, we don't pronounce it boracic, it is pronounced as brassic, possibly due to people thinking that there is a link to the word brass, considering the financial connotations. (brass is a slang term for money, particularly coinage - reinforced by the proverb "Where there's muck, there's brass.")(Wikipedia)
china -> china plate -> mate
This one needs little explanation, we had china plates coming in to the capital by the boat load, it rhymes nicely with mate, which is the most popular word for friend in most London areas. Though bruv gets used a lot these days in its place.
cobblers -> cobbler's awls -> balls
Balls being used in the slightly vulgar sense that bollocks is usually roped in for. To say that something is untrue, or rubbish in some respect, mainly refuting something said by another. The primary phrase I hear it used within is "What a load of old cobblers!"
mickey -> Mickey Bliss -> piss
Mickey Bliss was a bloke, supposedly from London, who no one knows anything about - he remains unidentified. Except in infamy for being the source of this rhyming slang couplet, quite a recent edition to the pantheon, 1930s I believe. Always used in the phrase "taking the mickey", although michael/mick can be substituted, from "taking the piss", which we recently had a discussion about. Note you can not use mickey for the act of urinating, the rhyming slang for that is jimmy (from Jimmy Riddle -> piddle).(Wikipedia)
scarper -> Scapa Flow -> go
The etymology on this one is a little unclear, it is known as rhyming slang from Scapa Flow (a very important natural harbour in the Orkney Islands) - but came additionally (and most likely originally) from the Italian immigrant population via their verb scappare which means to escape. The current meaning is to make a quick get away, we had a discussion on this word recently. It was used heavily amongst certain groups, notably it was taken up in the Polari language in the old phrase "scarper the letty", letty meaning bed/board/lodgings in Polari.
butchers -> butcher's hook -> look
We say, "take/have a butchers" to mean a quick look, synonymous with the word gander when used in that sense.
Hope that is of some interest to some of you, I do love etymology, particularly for obscure colloquialisms. :-)