I mean the term as in “Stop bugging me”.
- Did the term originate from the British term “buggering”?
- Is the term “bugging” used in that sense in Britain, Canada, and Australia?
I am quite sure there is no connection. “Bugging” is a synonym for “irritating” which indicates to me irritating bugs like mosquitoes and flies.
Think of a bug swarming around you… you swat but it won’t go away. It is really “bugging” you.
- It’s used in that sense in Australia.
Which naturally leads to the question of where bugger comes from.
The Online Etymology Dictionary is your friend.
Bugging as in bothering is surprisingly recent, according to that page:
Wait, I think I misunderstood you question, #2 is a bit ambiguous.
“Stop bugging me” is used in Australia in the same sense as in the USA. It’s not used in a way that is related to the word “bugger”, which is commonly used mainly as an exclamation.
I’m amazed to see that the use of the word bug to mean recording something with a concealed microphone dates all the way back to 1919.
Any idea when bug first began being used as a slang term for a crazy person? I remember it from back in the late sixties.
“Stop bugging me” and “stop buggering me” have two fairly distinct meanings, I’d say.
Who doesn’t like a good buggering?
Why are you buggering us about this?
Anyway, it is used in Britain, but was probably imported from America.
Don’t you think you’re being a little anal about this? 
Hey, man, get off my ass!
The OP may not be aware that a more recent extension of the transitive “bothering” meaning is an intransitive sense which means “freaking out” or “reacting angrily”.
"Why are you bugging? It’s no big deal."
It sounds California-surfer-1980s to my ears, but could have black urban origins, like its near-synoym* trippin’*.
Oh, bugger!
In Canada, the term is used synonymously with “bothering” or “irritating”; that is, “quit bugging me” means the same as “quit bothering me.”
Loved that ad.
When it first came out there was a big hooha (not in the vaginal sense, in the freaking out of the pubic sense) and it went to the Broadcasting Standards Authority who decreed that “Bugger” has changed it’s meaning so much from the initial sodomy root (heh) that it can’t be considered a swear word. Much to the joy of every schoolchild in the country.
Buggerknuckles
Wouldn’t it be more like"Stohuh huh huh huh buh buh buh gering me uh uh uh "?
My own personal hypothesis is that it derives from the usage of the word ‘pest’. As in, “You are being a pest”. From which comes, “Stop pestering me”. Hence the progression to “bugging”. I could be wrong.
Rojelio, it doesn’t even need to be that direct–it could just be a linguistic case of convergent evolution, rather than people deliberately substituting a synonym.