Green’s Dictionary of Slang

wobbly n.2

also wobbler

a fit of nerves, of panic, of bad temper; thus one who has such attacks.

[US]Ersine Und. and Prison Sl. 79: wobbler, n. A wingding, ding.
[US]Goldin et al. DAUL 238/2: Wobbler. A wing-ding beggar, that is, one who throws wing-dings or fits to engage sympathy.
[US]J.E. Schmidt Narcotics Lingo and Lore 185: Turn on the wobbler – Of a drug addict, to feign an illness, fit, or spasm.
[UK]K. Lette Llama Parlour 184: Abe paid off everyone imaginable to keep Pierce’s ‘wobbly’ out of the papers.
[UK]Guardian G2 23 June 2: She had not finished her impression of a two-year-old in the midst of a lost-dummy wobbly.
[UK]K. Richards Life 188: We’d already had a wobbler with the formidable but brittle Andrew Oldham.

In phrases

chuck a wobbly (v.)

1. of people, to panic, to suffer a fit of nerves; also used fig. of things.

[Aus]T. Winton That Eye, The Sky 84: Dad talks funny and chucks a wobbly in the bush.
[UK]K. Lette Foetal Attraction (1994) 152: Listen, sorry I chucked a wobbly.
[NZ]McGill Reed Dict. of N.Z. Sl. 232: wobbly A tantrum. Often used in the phrase chuck /throw a wobbly. From mid 1960s here, elsewhere later.

2. (Aus./N.Z.) to tell a dubious story.

[Aus]Macquarie Dict. [Internet] chuck verb 1. to do or perform right away: chuck a U-ie; chuck a mental; chuck a wobbly.
throw a wobbly (v.) (also throw a wobbler)

of people, to panic, to suffer a fit of nerves; also used fig. of things.

[UK]A. Bleasdale Scully 13: The Parish Priest threw a wobbler [... ] an’ pulled me off-stage.
[Aus]Teleg. (Brisbane) 13 Apr. 13/2: The world has used him, exploited his talent and brains and then discarded him…is it any wonder that he throws a wobbly?
[UK]T. McClenaghan Submariners II ii: I can’t throw a wobbly in the middle of a situation like this.
[UK]A. Bleasdale No Surrender 74: They’re throwin’ wobblers at the next table.
[UK]F. Taylor Auf Wiedersehen Pet Two 30: That tattoo you had. With the other bird’s name on. That must have thrown a bit of a wobbly.
[Ire]R. Doyle Snapper 2: I should give ou’, I suppose. An’ throw a wobbler or somethin’.
[UK]Observer Business 25 July 16: Some of them will completely lose it and throw a wobbly.
[UK]J.J. Connolly Layer Cake 249: They would argue about shares, percentages, minute tiny details, throwing wobblers, crying.
[UK]J. Cameron Hell on Hoe Street 60: Teach them some fucking manners, throw a fucking wobbly soon as they rattle their cage.