I think the earliest walk-through enclosures would be pens for pinioned waterfowl, like the ones at Slimbridge and the other WWT centres visitors, which must be older than the earliest zoo: according to Wikipedia there have been pelicans in St James's Park since 1664. I would imagine that another of the early types of zoo walk-through would be the 'Pets Corner' type of exhibit, based on a farmyard.
However in terms of a proper modern zoo exhibit, the Tropical House at Chester (now called the Tropical Realm) was the first walk-through with free-flying birds that I saw. In answer to a question about the opening of the Tropical House at Chester in another thread, bongorob provided the date:-
13th June 1964, which was the 30th Anniversary of the founding of the North of England Zoological society.
I'm not sure when the Tropical House at Slimbridge was built, but that may have be of similar age. However I think that the principle was established previously in the free-flight hall of the Bird House in Frankfurt in 1962, and there may be earlier examples in Europe or the USA.
The oldest walk-through outdoor aviary in the UK is said to be the Snowdon Aviary at Regent's Park, which was opened in 1965.
The first British walk-through mammal enclosure that I remember is the Lemur Wood at Blackpool which was opened in 1999, but I suspect that there may be older examples in the UK and there certainly are in Europe. I remember walking, rather nervously, through the colony of Tonkean macaques at Thoiry near Paris in 1972.