This idiom began to be used generally in the English-speaking world from its use as the title of a film (1997) about a group of unemployed British factory workers who raise money by staging a strip act (taking off all their clothes) at a local club.
The origin of the expression is uncertain. In Brewer's Dictionary of Phrase and Fable, first published in 1870, but revised by Adrian Room (Cassell, 2000), we find the following explanation: "It may derive from ...the Spanish card game 'monte' (literally 'mountain', i.e a heap of cards)...or from Field Marshal Bernard Montmogery, nicknamed 'monty' (1887-1976), said to have begun everyday with a full English breakfast when campaigning in the African desert in the Second World War."
"Do you mean they take off all their clothes?" "Yes, they go all the way, the full monty."
The wedding reception was magnificent; there was champagne, caviare, a three-couse meal, two bands - the full monty.