In a previous post about the “Peppa Pig Effect” — the phenomenon of American toddlers picking up British terminology and accent from a television cartoon — I mentioned a Guardian article on the subject whose title starts “Having a Go.”
“Have a go” is indeed a Not One-Off Britishism, of the historical variety (meaning that its U.S. adoption wasn’t recent). Its original meaning, says the OED, was “To aim a blow or shot at someone or something; to make an attack or onslaught upon someone or something.” The first citation is from Lady’s Magazine in 1792: “I felt such a flow of spirits and courage, that I hid myself behind a tree, determined to have a go at him—the moment he passed me, I fired my pistol.”
Fairly quickly it took on a broader sense of attacking verbally or criticizing, and the still broader sense (with which I associate the phrase) of “To make an attempt at something; to have a spell or turn of doing something.” An 1863 citation is from Charles Reade’s novel Hard Cash: “You have stumbled on a passage you can’t construe… Here, let me have a go at it.”
It and (at first glance) all the other citations for all three senses are from British sources. However, Green’s Dictionary of Slang, interestingly, has African-American sources for the attacking meaning, including John Ridley’s 2002 book A Conversation with the Mann: “He was fixing to have a go at these boys like he was Charlie Bad-Brother.”
And I can attest that the meaning of having a try or a turn has been widely used in the U.S. for some time. For example, a New York Times review of the 1961 film Voyage to the Bottom of the Sea has the line, “There’s no earthly or oceanic reason why [director Irwin] Allen shouldn’t have had a go at such a subject on a frank entertainment level.” And in a 2020 Times article about the television program Dispatches from Elsewhere, the (American) reporter writes of an actor, “He had a go at defining the show.”
The Guardian article (as opposed to the headline) says American Peppa Pig watchers have adopted a rather different phrase, “give it a go.” To my ears this means basically the same thing as “have a go.” Neither the OED not Green’s has an entry for it, but it shows up in two OED citations in a definition for “go” (as try or attempt). The second is a recent quote from an English newspaper but the first, surprisingly is from The Boston Daily Globe in 1892: “There was an air of diffidence about the different drug stores that were opened yesterday, which plainly said, ‘We are not sure of this matter, but we’ll give it a go and see how it comes out.’”
That would appear to be an outlier both linguistically and geographically. Searching the vast Google Books database, I don’t find “give it a go” showing up till 1915 or ’16, with the early uses all being from Commonwealth sources, especially Australia and New Zealand, as in this screen shot:
And Ngram Viewer, which is based on Google Books, shows that both phrases began in Britain (the blue and red lines), then gained popularity in the U.S. (orange and green) in the 2000s — which is, of course, the era of NOOBs.
Ngram Viewer also confirms my guess that “give it a shot” is the American version of the synonymous “give it a go.”
notoneoffbritishisms.com 
The expression “have a go” received an enormous boost in the UK from a radio program(me) of that name that was broadcast from 1946 to 1967. Very popular; at one time getting 20 million listeners. See
https://en.wikipedia.org/wiki/Have_A_Go
It had its own song, “Have a go, Joe,” the words of which can be found on the site below.
http://whirligig-tv.co.uk/radio/haveago.htm
The host was a Yorkshireman named Wilfred Pickles. In the early days of WW2 he became a news reader for BBC radio – the first person with a regional dialect to have such a post I think. This was widely taken to be an attempt to gain a more inclusive feel for the BBC and thus to cement national unity, but I think that it was later discovered to be because it was reckoned that in the event of an invasion the Germans would be unable to replicate his dialect and mislead listeners. The 1957 British film “The Naked Truth” (“Your Past is Showing” in the US, where it was felt that the word “naked” would encourage false expectations I suppose) had Peter Sellers playing a character widely believed to be based on Pickles: folksy on air but a so-and-so off. The character (Sonny MacGregor) was Scottish, not a Yorkshireman – presumably to dodge the libel laws. It’s a very funny film!
‘Have a go’ was also the name for a BBC radio quiz show. This is from Wikipedia:
Have A Go was a BBC Radio show that ran from 1946 to 1967. Hosted by Wilfred Pickles and co-presented with his wife Mabel (nee Myerscough), it involved the couple travelling to venues around the UK and speaking to members of the public, who were then invited to answer quiz questions in the hope of winning a small amount of money. It was the first quiz show in Britain to offer such a prize.[1]
Pickles’ presentation style resulted in catchphrases such as “How do, how are yer?”, “Are yer courting?”, “What’s on the table, Mabel?” and “Give him the money, Mabel” or “Give him the money, Barney”.[2] The programme’s popularity was such that at one time it was attracting an estimated 20 million listeners weekly. For six years the pianist for the programme was Violet Carson, later to be famous for playing Ena Sharples in the long-running TV soap opera, Coronation Street. [3]
Have a go meaning what you write or, similarly, “take a turn” is pretty close, but Brits also use it to mean “make fun of,” which, thankfully, Americans don’t appear to have been persuaded to have a go at much in speech.
Growing up in the late sixties and early seventies as a Londoner (with an Irish mother), “have a go” was mostly used in the confrontation sense, whilst “give it a go” always was the same as “have a try at it”.
“Instead of having a go at me, why not address my comment?”
“Some say we should be more empathic when responding to strangers. I’ll give it a go.”
(Yeah. I said ’empathic’. Although living in the States, I am yet to be acclimatized to the vagaries of their dialect!)
In Australian English (probably derived from UK) , “Have a go, you mug!” can be a taunt, an invitation to a fight.
From the 1960s onward members of the public who tackled a bank robber or other felon were described by the Press as “Have a go heroes”
I love that.