What do you mean by accreditation? If, re the UK, you mean having a Zoo Licence than I think the list will be very short (i.e. near nil). The only "zoos" that occasionally don't have a licence are expanding "glorified farm parks" who "don't know the rules".
It depends what you mean by accredited. Most UK zoos and certainly all the major ones are not accredited for there EAZA membership. The accreditation and inspection process was not required when they first joined. Only more recent members have been accredited, which is now a condition of membership. The first EAZA accredited zoo in the UK was the Shaldon Wildlife Trust back in the 90s. However most UK zoos are accredited members of BIAZA.
In North America there are around 220 AZA accredited facilities, meaning that the Association of Zoos and Aquariums rigorously inspects each establishment at least once every 5 years. Only about 10% of all wildlife facilities are AZA accredited, and in my opinion even some of those have more than their fair share of dodgy animal exhibits. Some folks think that there are some great little zoos that are not AZA accredited, but I'd be shocked if they could name 5 that are even half-decent. For whatever it is worth, AZA accreditation in North America is a huge deal and institutions without it are for the most part substandard "homegrown" zoos, research facilities, etc.
I don't think Monkey Jungle in Miami, Florida is accredited, is it? Generally AZA zoos are better-ran than non AZA zoos but there can be exceptions.
I'm quite curious to see if anyone can come up with any zoo that is half decent but is NOT accredited. Here is the link to the AZA accreditation page, with many more links on the actual site itself: What is Accreditation - Association of Zoos and Aquariums
An interesting place is Reptile Gardens in Rapid City, SD. It is not AZA accredited, but is half decent. They have Komodo dragons, aldabra and galapagos tortoises, and a fifteen foot long saltwater crocodile. THe grounds are beautiful, and the education elepment is very well done. Wild Animal Park | South Dakota Attraction | Reptile Gardens
I currently work for a non-AZA accredited facility. Even though we do not have accreditation, we hold ourselves to the highest standards of animal care and welfare. We are a fairly new facility and will probably apply for accreditation after we complete the new facility.
I can think of several right of the top of my head, mostly reptile related facilities- its my primary interest: Gatorland, Living Planet, Reptile Gardens, Clyde Peelings place, Crocodile Encounter, numerous drive through parks some of which are really excellent. Some of these may be ZAA. The AZA cert is expensive and some might add simply not needed. It can cost upwards of $10000 for the cert which is simply absurd. On the flip side a private zoo is to a large degree only as good as the standards of the owner. Some likely exceed what AZA requires others may not.
There are at least two good zoos in California that are not AZA accredited, both focus on non-releasable native wildlife and rescued animals. The Orange County Zoo has some excellent exhibits for mountain lions, black bears, and the only beaver exhibit in California (Orange County California - OC Parks). The Folsom City Zoo similarly has excellent mountain lion and bear exhibits for animals that were either injured or "problem animals" that would have otherwise been destroyed. They also have wolves and many other native species, along with some rescue tigers and monkeys (Folsom Website - Zoo Sanctuary). From the photos in the gallery it looks like the California Living Museum in Bakersfield is a pretty high-quality non-AZA facility, but I have not been there.
I know Clyde Peelings place is aza. I am a former keeper that worked at Gatorland. When I was there I know that the keepers wanted aza, however, since it was a family owned business they did not want to 1. go non-profit 2. incur the fee's associated with membership. I know there's probably more to that, but that was the reasoning given to us. It's still a great facility, even though they're not accredited.
Toronto Zoo has recently lost its AZA membership because of politicians nosing in business they know nothing about :/ I still think it is a great zoo. Eurasia (a out of date section of the zoo) is being renovated and their breeding programs are something to be proud of. thestar.com iPhone : Toronto Zoo?s ?hidden work? is saving endangered species
It was over the issue of the elephants being sent to PAWS (the city overriding the zoo board): Toronto Zoo | AZA Accreditation Denied to Toronto Zoo for 2012-13
I just found this list of non-accredited zoos in the U.S.: http://www.zoology.msu.edu/uploads/documents/nonAZAaccredited2010.pdf