I have covered the use of BrE “bits” instead of the AmE “parts,” as in “the good bits,” “the naughty bits,” “lady bits,” “dangly bits,” etc. I’ve recently noticed another “bit” popular among journalists, as a synonym for a piece of work, traditionally and customarily shortened to a “piece.” David Carr of the New York Times is fond of it, and here it is from Michelle Dean on Twitter: 
The OED suggests that the use of bit by itself in this way derives from tit-bits or tidbits, referring to a number or series of small items. The dictionary gives these citations:
1896 Daily News 4 Nov. 2/7 This is a weekly journal called ‘Gems’. As its title suggests, the new paper will be of the ‘bits’ order.
1928 Granta 30 Nov. 172/1 If the editor of the Review were to ask me to write a little bit about Christmas I should laugh in his face.
The newfound popularity may stem from the fact that the flood of communications we are flooded with in this day and age, any one of them, no matter how long, starts to seem like a tidbit. Or, in fact a “bit”–in the sense of the tiny pieces of information by which computers operate. (The word, which dates from about 1947, was coined by J.W. Tukey as a combination of “binary digit.”)
But all that is a bit of a speculation. No pun intended.
notoneoffbritishisms.com 
My British friend Barbara refers to tartans as “gingham with fussy bits”, to which we, American weavers who love the Scottish Games, just smile.
I have heard…often, expressions with “bit” used for emphasis.
eg. “He went on a bit during the presentation.” He went on speaking far too long.
” that is a bit harsh.” That is worse than harsh
” she is a bit of all right .” Meaning a very attractive female, in this case.
Is this what you mean?
Arthur, numbers one and two are common in American English. I’m aware of number three as a Britishism, but have not spotted it over here
I understand.
As you will probably know, in England the expression “I wrote a bit for the professional journal” could be interpreted as ” I wrote a lot” or ” I wrote one article”
I find these nuances quite interesting. I sat watching TV here in Prince Edward Island last night and heard “bit” twice. Both occasions the expression was “a little bit” meaning ” a little bit”
And I do realize this is Canada and not the U.S.!
In our family (Quebec and Ontario, Canada) we may say something about getting or giving “the best bits” when serving up a meal. It occurs to me we also talk about “the good bits” or “my favourite bits” of a book or movie.
And of course there’s the song from My Fair Lady – “With a little bit of o’ luck”. I’ve no idea if that a “Britishism” or not??
Only if you pronounce it with a glottal stop!
Here in the UK the word ‘bit’ seems to have an almost infinite range of uses, and we would probably be in a bit of a mess without it. Apart from all those already mentioned, it can also mean a lot, as in ‘I bought quite a bit today’ and ‘it cost me a fair old bit’. It can also mean a small amount, as in ‘More pie?’ – Just a bit, please’. The removeable sharp bits that fit into the chuck of a drill are called ‘drill bits’. If you’re stressed you might ask for ‘a bit of peace’, which could happen if someone has been ‘going on a bit’ about something that ‘doesn’t make a bit of sense’.
A bit like the bit I’ve just written, perhaps…
Bit seems to be almost as useful as “get”. A Swedish friend of mine commented that yu could use “get” to describe your day. Got to bed. Got up. Got a coffee. Got to go. et cetera