The slang phrase like a bride’s nightie jocularly alludes to the quickness with which a bride’s nightdress comes off on the wedding night.

It is used in:
– to be, also to go, etc., off like a bride’s nightie, meaning to make a hasty departure, also to start quickly and make good progress;
– to be, also to go, etc., up and down like a bride’s nightie, meaning to rise and fall quickly or repeatedly, also to fluctuate rapidly.

With one exception, all the earliest uses of like a bride’s nightie that I have found are Australian.

The earliest is from Blossom Like a Rose (New York: Viking Press, 1969), a novel by the English-born Australian author Christopher Bray (born 1938):

Brody gave him a playful punch in the ribs and he ran around the truck and jumped up into the cab.
‘Come on youse blokes!’ he shouted. ‘We’re off like a bride’s nightie!’

The second-earliest use of like a bride’s nightie is from Jockey Rides Honest Race (Sydney: Ure Smith, 1972), by Geoff Morley:

At the furlong the jockey said ‘Let’s go’, and Che Sera went off like a bride’s nightie. In a flash she hit the front and won by several lengths.
—as quoted by Gerald Alfred Wilkes (born 1927), professor of Australian literature at the University of Sydney, in A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990).

The only non-Australian early use of like a bride’s nightie is from Murphy Sees Plot on Shelf Oil, published in the Daily News (New York City, N.Y., USA) of Sunday 29^th June 1975, in which Jeffrey Antevil quoted John Michael Murphy (1926-2015), a member of the United States House of Representatives, as using the phrase:

Washington, June 28 (News Bureau)—The United States Geologic Survey’s recent drastic downgrading of its estimates of oil resources off America’s shores may have been a political ploy to discourage Congress from passing legislation to regulate development of the outer continental shelf, according to Rep. John Murphy (D-N.Y.).
[…]
“They’re trying to make us believe there’s not enough there to legislate about,” Murphy said, declaring that the administration’s reserve projections were “going up and down like a bride’s nightie.”

Kevin Childs wrote the following in his review of The Australian Slanguage: A Look at What We Say and How We Say It (Sydney: Cassell Australia, 1980), by Bill Hornadge—review published in The Age (Melbourne, Victoria, Australia) of Saturday 27^th September 1980:

Until a few years ago, only insiders knew that the injunction “bums to mums” was used by footy coaches to warn their players that sex on the eve of a game might sap energy. Yet another term to emerge is that used up Cairns way for young ladies from the south who may appear on the decks of the fishing fleet. “Marine biologists” is their particular euphemism.
In a similar vein is that great term for a fast mover: “Off like a bride’s nightie.”

John Douglas Pringle wrote the following in his review of Eric Partridge in His Own Words (London: Andre Deutsch Ltd, 1980), by Eric Partridge¹ and David Crystal²—review published in The Sydney Morning Herald (Sydney, New South Wales, Australia) of Saturday 24^th January 1981:

Are Australians, then, a nation of poets? Well, not quite. The trouble is, as Partridge pointed out, that while slang is originally invented by individuals of wit and ingenuity, it is quickly adopted by the mob and repeated so often that it loses all freshness and meaning. The first Australian who said that something came off “like a bride’s nightie” was a poet. The thousandth was just a parrot.

¹ Eric Honeywood Partridge (1894-1979) was a New Zealand-born British lexicographer.
² David Crystal (born 1941) is a British linguist, academic and author.

Eric Partridge, by Pascual Locanto—from The Sydney Morning Herald (Sydney, New South Wales, Australia) of Saturday 24^th January 1981: