Chucking a wobbly comes from throw a wobbly and wobbly refers to a fit of anger, possibly suggesting the person is mentally unbalanced.
The 1994 Shorter Slang Dictionary (Partridge, Beale, Fergusson) says:
throw a wobbly to become angry, agitated or mentally unbalanced; to behave irrationally or unpredictably. Later 20th century.
The 2008 New Partridge Dictionary of Slang and Unconventional English says:
wobbly noun a fit of anger UK, 1977. < throw a wobbly to have a fit
of bad temper or anger NEW ZEALAND, 1984
The OED says:
to throw a wobbly : to lose one's self-control in a fit of nerves, panic, temperament, annoyance, or the like; also, to act in an unexpected way, causing surprise or consternation.
Their first quotation is from the Brisbane Telegraph on 13 April 1977.
I found the phrase slightly earlier in the Australian Trove archive of newspapers in The Canberra Times of 12 March 1977:
The story begins with
Howard Beale, a
commentator, throwing a
wobbly on camera and
It shows up in a 1964 Google Books snippet of The Watersiders by Michael Davis:
They were laying down long planks at odd angles from the square as though about to participate in some peculiar rite. "What's the matter, mate? You look queer. Ain't going to throw a wobbly, are you?"
Google Books snippets are sometimes wrong, but the author, title and year seem correct and it was published in Australia so it's plausible.
Another Google Books snippet is from the 1972 Parliamentary Debates of New Zealand:
Mr Moyle — Perhaps he will throw a plum at them. Mr TIZARD — He can throw a plum, or he can throw a "wobbly", which is more in keeping with the Minister. He complained that the Opposition did not congratulate the Government and the ...
Searching for 1972 brings up correct-looking results. Furthermore, Colin Moyle was an MP from 1963-1977 (and 1981-1990), as was Bob Tizard from (1957-1960 and) 1963-1990.