rather than just simple escapes, this thread may be more interesting for species that have actually established wild populations from zoo escapes. The following is mainly gleaned from Christopher Lever's "Naturalized Mammals of the World" (1985)...
Red-necked (Bennet's) wallaby: in the UK, from a private menagerie near Leek in Staffordshire, escaped in 1939 or 1940 (5 animals) and formed the population that survived until recently (or does still) in the Peak District National Park. A second colony that formed in Sussex was presumed to have originated from Leonardslee Park near Horsham.
Brush-tailed rock wallaby: in Hawaii, one pair escaped from a private menagerie in August 1916. The resulting population is still there today.
Grey kangaroo: in France, resulting from a bungled burglary attempt of a zoo (see the Grey Kangaroos In France thread)
Taiwan macaque: in Japan, established on Oshima after escaping from a zoo during WW2
Anubis baboon: in Spain, 60 baboons escaped from the Auto Safari Andaluz safari park in Cadiz in 1972. Forty were soon killed by hunters but the rest survived unnoticed until their resulting population was "rediscovered" in 1992!
Chinese water deer: in the UK, some escaped from Woburn during WW1, established themselves in the surrounding areas, and were a few years later supplemented by escapes from Whipsnade. Now common in the UK.
Reeve's muntjac: in the UK, established following escapes from Woburn and Whipsnade. Now common in the UK.
Indian muntjac: in the UK, established following escapes from Woburn and Whipsnade, but no longer occurring wild there
Himalayan tahr: in South Africa, in 1937 some (possibly a pair) escaped from the National Zoo in Pretoria and established a population on Table Mountain, where they still occur.
In the USA, Argentina, etc there are numerous large ranches that hold free-roaming ungulates, especially antelope species, and several of these have resulted in escaped animals forming wild populations
Indian palm squirrels: in Australia, in about 1942 some escaped from Taronga Zoo and established a population in the surounding area. They are no longer found wild there, but do still live wild around Perth Zoo, where they were deliberately released in 1898, and from where the Taronga ones were obtained.
Taiwan golden-backed squirrel: in Japan, escaped from a zoo on Oshima during WW2. By 1950 there was a population of 20,000.
Siberian chipmunks: established from escapes from captivity in France, Germany, the Netherlands, and Austria
Black-tailed prairie dog: in the UK, "after a number escaped from a country park on the Isle of Wight in southern England when it closed down, they rapidly multiplied, yielding 250 - 400 specimens by the early 1990s. Attempts to snare them have largely failed, as they are very intelligent, and it is feared this New World interloper may overrun the entire island" (from Karl Shuker's Mysteries of Planet Earth, 1999)
Himalayan (Hodgson's) porcupine: in the UK, a pair escaped from the Pine Valley Wildlife Park, Devon, in 1969. Several were killed between 1971 and 1975. The population was either eliminated or died out.
African crested porcupine: in the UK, a pair escaped from their cage in the botanic gardens at Alton Towers, Staffordshire, in 1972. This species was likewise only found in the wild in the UK for a few years before either being eliminated or dying out.