can British people tell different english accents?

Mostly but not all the time. I moved to Scotland from England and a friend from Essex came to visit. My Scottish friends thought he was Australian .  

Of course, often to within a few miles.

I'm American and I can tell different UK accents. 

My mom is a scouser from Liverpool, England.. she sounds very different to people from London. 

Yes.  And different Irish accents very often.   Accents can change every 25 miles in the UK.

YES. Funny story. I moved from the UK to the US in the early 90's and then in 2000, I was at the Sydney Olympics. I heard a family on the morning train out to the Olympic main venue. Some of the family were local and some from the UK. And I could NOT tell the difference between British and Australians accents. I heard that said about Americans, before I left the UK, but it's true !!!

Yep, people from the same country and even from the same city can speak with different accents. That could be a fun game guessing who is from where ))

I read recently that Brenda Glethin, who plays 'Vera' in the detective series of that name, is not from the north-east.  If that is true then she is a consummate actor, I had no notion by voice that she was not a local.  (But I am a southerner, so how should I know?!)

Yes i can tell Most Starting with Estuary to Welsh Devon and Cornish to Scottish Lowlands to Highlands

everybody recognizes the Liverpool and Yorkshire Accent

Mary Poppins Ruined by some idiot who could not speak English

Yeah, most British people will be able to identify a range of accents (and tell if they're being done badly; eg Dick Van Dyke's character in Mary Poppins).

There's not a single London accent, so this example doesn't really work. 

Many British actors can do excellent regional accents. 

It is difficult nowadays, because of the influence of television. Watching Youtube it is hard to tell what country people are broadcasting from as many seem to have a strange sort of mid Atlantic accent. Somebody like David Tennant does a good English accent, despite speaking with a strong Scots accent off screen; you would not know Judi Dench was from Yorkshire; but most production companies have someone to help coach the actors. It is far easier to tell a foreigner speaking English, even if they speak with a near perfect accent.