I think this is in reference to the past perfect tense specifically. So "I have gotten sick" vs "I have got sick". I don't know if the latter is actually used in British English, but it's my understanding that the former would never be used since it's grammatically incorrect in that dialect.

You have got to be kidding me!

Neither dialect says you have gotten to be kidding me, right?

Its not formal english though. The British informal equivalent would be 'you're having me on!' or 'you're having a laugh!' or (rude) 'you're taking the piss'