Noun. Part, as of a text or film, usually used in the plural. “He [Kenneth Starr] wants America to believe he’d only included the good bits to help the legislature reach an informed decision.” (Time Magazine, August 9, 1999)/”I can tell you some of my very favorite bits. Every single bit of the fight with Matthew Patel is brilliant.” (LubbockOnline.com, January 12, 2011)
notoneoffbritishisms.com 
As a Canadian who spent years living in Britain and was married to an Englishman, this is … interesting. Where I live many of these terms are the normal way of speaking. I had no idea that saying “what is your dog called” was British, for instance. Most of these are daily use here, though the term “in future” still grates as it tends to be said with a degree of pomposity not often achievable by “colonials”.
Anna, the “called” thing is specifically referring to humans, as in “I have a friend called Charles.” Americans would not say that.
As a middle-aged Canadian with literary (I typed ‘loiterary’ which is, sadly, far more accurate) interests & dead brit parents, I find all these terms in my passive lexicon*, and about three quarters in active. I might not have the fabled ‘mid-atlantic’ accent, but without a doubt I have a mid-atlantic vocabulary.
But as far as I’m concerned, linguistic flexibility is an ability I’m proud of, not one I find annoying. I roll my eyes at speakers who use words wrongly, or who don’t entirely understand the word they’re using, but respect people who can shift registers at will, varying them for the audience or for (most nobly comic) effect. Vastness of vocabulary is the english language’s greatest strength and most salient characteristic. Why not rejoice in it?
*Passive is what you understand, active is what you use.
TurboTax, while it’s loading, “Hold on, we’re getting all the technical bits together.”
Of course, “bits” in British English is also downmarket slang for genitalia – “he was staring at my bits while I was showering!”…. Short for “bits and pieces” I would guess.
The more I read this wonderful blog, the more I realise that every single word in British English is a form of slang, euphemism or synonym for genitalia. Or if not for the actual ‘bits’, then for what you might do with them. :SIGH:
Also with the “called” thing. A person might me named Maurice but called “Mo”.
While working as an expat at a British turbine blade manufacturer, I was puzzled to hear the Brits use the term “bits” when referring to blades, as in, “Where are the bits now?” In our domestic factories we always talked about how many “pieces.”
I’m from Lancashire and we would ask “What is he called?” or “What is his name?”, however where I live in West Yorkshire, they would say “What do they call him?”. To me, absurd, but I quite like it.
In British English “bits” is also used for the thing in orange juice that Americans call “pulp”. Here is an example: http://www.tesco.com/groceries/Product/Details/?id=256441348.
We call it “pulp” too, but “bits” are the miscellanous “things” in it….
For the difference between naming and calling, consult Lewis Carroll! http://www.haddockseyes.com/
“bits” simply means miscellaneous non specific items, I’m astonished its not known over there, it just seems so… ordinary. Also the piece that goes in a horses mouth is called a “bit” as well sometimes referring to genitalia, presumably short for “dangly bits”. This is sometimes combined to comic effect as in “the horses bits were foaming”
To clarify further “bits” is similar to, but not the same, as “parts”. For example “I ordered new parts for the engine” but compare to “after assembling the furniture from Ikea there were lots of bits left over”. “Parts” is specific, a known quantity, “bits” is non-specific, vague, unknown.
The Pythons often used to refer to “naughty bits”.
Only in the sketch “How to recognize different parts of the body”, which included “Number five. The naughty bits.” “Number seven. Two inches to the right of a very naughty bit indeed.” “Number eleven. More naughty bits.” “Number twelve. The naughty bits of a lady.” “Number thirteen. The naughty bits of a horse.” “Number fourteen. The naughty bits of an ant.” “Number fifteen. The naughty bits of Reginald Maudling.” “Number twenty-seven. More naughty bits.” “Number twenty-eight. The naughty bits of the Cabinet.”
The phrase “bits and pieces” is common in the US. Bits meaning small parts is also common (as in the Time quote abuv). It’s also part of the tech lingo. There may be an oddball use of it in the UK that’s not found in the US but overall I don’t see how you even think “bits” is a Briticism.