None of the Italian Job Mini's were modified to 4wd that i know of.....but as said above this is the usual explanation:
"When the Coopers are being prepped in the garage prior to the Job, one car has the rear end jacked up. The mechanic says something is wrong with the differential. Swapping the diff in a Mini (a front-wheel drive car) requires the removal of the engine and gearbox which are at the front, not the back. However, this could be an intentional 'mistake' left in by the writers/film-makers. When this film was made front wheel drive cars were not as commonplace in the UK as they are today and most mechanics where used to working on the standard rear wheel driven cars. Asking an inexperienced mechanic to change the diff on a Mini was apparently a common practical joke/jape. The film maker may have also been implying that one of Charlie's mechanics wasn't up to scratch as there are various other gags in the film referring to their ineptitude (i.e. "you're only supposed to blow the bloody doors off!")."
If you search on TMF there was however a 4WD BL Special Tuning Rallycross Mini Clubman in the 1970's with a rear diff.