If there is anyone here like me who is a little bit obsessed with service dog laws in other countries, here are the laws for the UK:

Service dogs (known as assistance dogs in the UK) are classed under the following 4 categories: 1. Guide dogs for the blind 2. Hearing dogs for the deaf 3. Dogs trained by one of the outlined organizations 4. Dogs trained to assist their handler with a disability Owner trained service dogs fall into the last category. This is as stated in the Equality Act 2010 (Link) under Part 12, Chapter 1, Section 173- Interpretation This is the section that describes what is defined as an assistance dog:

The only difference between an ADUK (charity) dog and an owner-trained dog is flying- Airplanes are regulated by a different set of laws, so they are able to deny non-organization dogs (to my understanding)

ADUK is a large charity that is comprised of several organizations that train assistance dogs. ADUK is a charity and is NOT associated with the government. Here is the ADUK website section about laws. Although it doesn't technically state that owner-trained dogs are not allowed, it does imply such (incorrectly so). They do promote only accredited dogs having access rights. 

This is from the ADUK FAQ page. 

This does not state that owner-trained and non-ADUK organization dogs have no PA rights.  Basically, what they are saying is that the organization will support you if there is a legal issue, but if you owner-train you are on your own. This is the same as pretty much anywhere as far as I know. It doesn't actually address public access, it just implies that ADUK dogs are the only ones that are capable of passing a PAT. Although ADUK acts as though they are law, they are just a charity organization. They are not law and have no influence on owner-trained dogs.