In a commencement address last month, former U.S. Secretary of State Rex Tillerson said, “When we as people, a free people, go wobbly on the truth even on what may seem the most trivial matters, we go wobbly on America.”
That “go wobbly” caught the ear of friend-of-NOOBS Stuart Semmel, as an echo of a famous Margaret Thatcher quote. She described in her memoirs a conversation with Pres. George H.W. Bush during the first Gulf War: “We must use our powers to stop Iraqi shipping,” she recalled telling him. “This was no time to go wobbly.”
The relevant OED definition is “Wavering, uncertain, or insecure; unreliable, unstable.” And I hasten to say that the word “wobbly” is nothing new in the U.S., especially in the literal sense of wavering back and forth. In addition, “Wobbly” is a slang term for a member of the International Workers of the World (I.W.W.) labor union. (A 1923 article by a member of the union gave an account of how the term originated: “In Vancouver, in 1911, we had a number of Chinese members and one restaurant keeper would trust any member for meals. He could not pronounce the letter w, but called it wobble, and would ask: ‘You I. Wobble Wobble?’ [that is, I. W. W.] and when the card was shown, credit was unlimited. Thereafter the laughing term among us was I. Wobbly Wobbly.”)
But “go wobbly” definitely has a Thatcherian and very British feel. Searching for the phrase in the New York Times, I found it in a January 2018 article about a conservative group’s hoped-for turn of events after a government shutdown a few years back:
“The public would express outrage that the president was willing to hold America’s full faith and credit hostage over the much-disliked Obamacare. Democrats would go wobbly.”
And then, not long after hearing from Stuart, I read in The New Yorker an article about the hacked e-mails of Sony Pictures co-chair Amy Pascal. One of them, from producer Scott Rudin, remarked, “So you’re feeling wobbly in the job right now.”
It’s clear: “wobbly” has arrived.
notoneoffbritishisms.com 
And you’ve also got ‘throw a wobbly’ for ‘throw a tantrum.’
As yet unheard in the USA I would say.
Do you have “throw your toys out of the pram” ? Similar meaning … not quite as severe as “go postal” …
We don’t even have prams.
Curiously, last night I was reading the story The Brave Little Toaster by Thomas M Disch and at one point, he mentions “a large vinyl perambulator, which is another word for pram, which is also know, in the appliances’ part of the world, as a baby buggy.” Quite why an American author writing for an American magazine, in a story quite clearly set in America, used the word pram, I cant tell.
“chuck a wobbly”
Every Whovian in the USA has heard wibbly wobbly timey wimey
“My spelling is wobbly. It’s good spelling but it wobbles, and the letters get in the wrong places.” -Winnie the Pooh
‘Throwing/chucking a wobbly or wobbler’ (tantrum) and ‘he was all wobblybob’ (a bit wonky, odd behaviour, erratic) are also British phrases.
Australians also throw or chuck a wobbly. Alternatively, they spit the dummy, or what Americans would call the pacifier. An epic tantrum would be called a dummy spit.
It seems likely that the origin of the phrase “ don’t go wobbly “ is a sexual metaphor .