Off like a bride's nightie
is a common Australian
colloquialism meaning "to do something fast".
The phrase wouldn't ordinarily be spoken in the company of ladies, although if it were, the recriminations from The War Office would be unlikely to be severe.
On a building site, you might overhear, "Jeez, mate, when he picked up his pay on Thursd'y arvo, he was off like a bride's nightie."