Green’s Dictionary of Slang

1. (N.Z.) one who prevaricates; thus personnified as Willy Wobbly, wobbling willie.

1905		Wanganui Herald (NZ) 1 Dec. 2: This might take with strangers, but not with the majority of electors, for we know that a man who changes his opinions many times is too much of the willy wobbler.
1906		Age (Melbourne)24 Oct. 11/4: When ‘wobbling Willies’ wobble for the last time [...] when the public refuse to engage men to make their laws at £300 per annum who would be overpaid at 15/- a week in a private business: then, and not till then, will the kingdom of clean sport be revitalised.
1912		Sthn Times (Bunbury, WA) 2 Nov. 3/4: The men must beware of the willy wobbler, as he is the most dangerous man to encounter.
1913		Internat. Socialist (Sydney) 30 Aug. 4/2: Willy wobblers who pose as Socialists and run as Labour councillors, who mouth Socialism from a Gaiety Theatre platform, and who tell the credulous to vote for the referendum.
1917		Lithgow Mercury (NSW) 19 Dec. 1/3: Mr. Dooley sarcastically described Mr. Hughes as ‘Willy Wobbly,’ akin to the thimble and pea man, ‘now you have him and now-you haven’t,’ owing to his continual change of front.

2. one who exposes his genitals in public.

1989

Mag. Fantasy & Science Fiction 128: Johnny had suggested that we could slip a few things into the system that would eventually find their way into a file on this new man in her life, and I'd said, Yeah, let's put him down as a willy-wobbler. Suspicion of indecent exposure wouldn't exactly be a career plus.

3. the penis.

2002		JethroTull.com Aug. 🌐 Pamela, nor have I probed the inner recesses of my psyche regarding the prospect of waving goodbye to Willy Wobbler and hanging the beef curtains, but [etc].
2009		E. James Breath of Fresh Air [ebook] ‘He hurt his willy wobbler as well,’ she added for good measure, when she finally bumped to a stop at the foot of the stairs.

4. a frightening experience.

2010

K. Smith Cowboys can Fly 125: ‘Should have grabbed it closer to the trunk.’ He gave it a good tug. ‘It’s still sound.' Cy laughed. ‘Phew. That was a willy wobbler.’ A second attempt, and Cy had made it to the branch.

5. on penis = fool pattern, an objectional person.

2016

www2.totalpolitics.com 9 July 🌐 At the slogan launch, Lord Mandelson told me David Cameron was ‘a willy-wobbler’. An eye-watering accusation. It later became clear he meant to say ‘wibbly-wobbler’, a reference to policy U-turns. Somehow, I prefer the imagery of the first version, although it's probably not one for the billboards.