I have always wondered if “Get away (with you)!” is a British way of saying “Come on!”? I mean, when you hear something silly, surprising, or simply unbelievable?
Do these two different expressions correspond more or less with each other?
Yes, you’ve got that right. One of those everyday, ideomatic expressions that can surprise you by not being universal.
I dunnow about “britons” in general, but two of my British coworkers do. One is from Newcastle and the other one from Leicester. The one from Leicester even doubles up: “oh cmon! Get away with you!”; this led to a lot of confusion the first few times, as some literally-minded foreigners weren’t sure if they had to go over to where she was sitting, go away, or what. The rest of the Brits in the team don’t use those particular expressions and I hadn’t heard them in Scotland either but most of the people with whom we had contact there spoke “business” English (that would be a ton of Scots with some Welsh sprinkling).
Your Geordie colleague might also use “Haddaway”, or in more relaxed circumstances “Haddaway and shite”, meaning “go away”.
If you mean, ‘come on’ in the sense of ‘advance’, then no. ‘Good heavens! I can scarcely believe the truth of the statement you have just made’ is approximately correct.
It has just occurred to me that a close British English colloquial equivalent to “Get away” would be “Come off it”. Just to confuse things 
Scots tend to drop the “Get” and just use “Away wi’ ye”.
Or, you know, American colloquialisms like “Get outta here!”
“Give over” is more weird, if you ask me.
Now that you mention it, I’ve heard “Come off it” as well. 
It can be used at different levels. If someone hears something is ludicrously implausible they might say ‘Getaway’ with a dry ironic tone. Somewhat like ‘You don’t say!’.
It would be used with more force ‘Get away with you!’ as an indignant admonishment for some impudence.
Down in London, they tend to say ‘Leave it out!’ but they often mangle the pronunciation with hideous glottal stops common in the Cockney dialect. ‘Leeeeve i aaaat’.
I just recently learned that “I’m across it” is there way of saying: “I’m on top of it”.
As in
“Lily, how are you coming along on those reports?”
“Yes guv’nor, I’m across it.”
Haven’t heard that one I must admit.
Sure it wasn’t “I’m all over it”?
Well, if you turn the closed caption on in Black Mirror, the phrase is said in the first episode and you can see it right there in the sub title.
I also heard it on another Brit show but I can’t remember which one.
Also:
Get Real! and No way are in common usage in the UK.
Leave it out! as mentioned is useful as a gentle rebuke. However it is tending to a rejection of what has just been said.
For the more forceful (and those who remember The Sweeney) Shut it!
Sadly some will use casual obscenities **Piss off! **or Fuck off! can be used (amicably) to indicate incredulity. They need to be used with wide eyes and raised eyebrows to avoid being too confrontational. On a gentler level Get stuffed! can be used.
Do what? is popular but can indicate a low IQ. Some favour the tautological rapid fire Do what? Do what? Do what? seeking a little comedy through repetition.
Vaguely popular after being used in a series of washing power adverts **You having a giraffe? **which derives from the more traditional (Are) You having a laugh?
And the standard, interrogative You serious? or You being serious?
TCMF-2L
I think that might be a news business thing. Certainly the first time I noticed it was journos using it on twiitter
Come off of it, to be precise.
I’ve never heard “across it” used like that, so I’d say it’s not in everyday usage.
I always associated the full “get away with you” as being Irish. It’s certainly rare to hear anybody in England say it in full, in my experience. From where I am from “give over” is most common, usually shortened to something approximating “giv o”.
You know, I think you’re right. The lady who said that phrase in Dark Mirror was indeed a journalist. (At least, if memory serves correctly)
Whereas in Ireland it’s not so much rare as completely unheard-of. To my Irish ears it sounds like something British, but obsolete.