there is no assistance dog register in the uk, as long as the dog mitigates the handlers disability, or in the case of a child the adult with them would be the handler.
there is legal requirement for an assistance dog to have an id book, like the ones you get from an aduk organisation. Owner trained assistance dogs are perfectly legal. There is also no legal requirement to have a vest/banada/harness or similar so the dog is easily recognised as an assistance dog.
so it would be illegal and disability discrimination to do some of the things other posters have suggested, such as a sign saying guide dogs and registered assistance dogs only.
there is no such thing as a registered assistance dog in the uk. There are aduk organisations that provide dogs to disabled people, but owner trained ads is also very much allowed. The aduk charities have a lot of requirements for there dogs (so many don’t fit the requirements despite being disabled etc), they have a limited amount of funds and dogs, and there’s not actually many aduk organisations. So this means many people do train their own assistance dogs, whether that be with a trainer, or a organisation that is not aduk, or just on their own.
the equality and human rights commission, states an assistance dog should not wander round the premises freely and should lie or sit quietly on the floor next to the handler.
if the dog is not doing this, ie bothering people, barking (unless the alert is a bark), generally misbehaving that sort of thing, you are well within your rights to ask them to leave. Obviously ads are not perfect so there may be little moments, so if it’s anymore than a little thing then you can ask them to leave.
service dogs in the uk are police dogs or similar. Emotional support animals are not a legal thing in the uk. The term assistance dog is the only legal term for a disability support dog in the uk. So if they are labelled anything other than assistance dog, then they may not be an assistance dog. Therapy dogs are a thing but they do not have legal rights of access in the uk if they are only a therapy dog.
the guide for businesses linked by @Reugny i was gonna post.
Also dogs for autism an aduk charity has this information https://dogsforautism.org.uk/ad-hi-information-page/#AD
if you suspect a dog is not an assistance dog you can ask the handler. But they are not required to tell you their disability, so I would stick to asking about the training of the dog, ie if it’s misbehaving question that rather than the fact it may or may not be an assistance dog. An assistance dog should be highly trained, over and above that of even a well trained pet dog. Though they are not infallible so I would bear that in mind too.