For the OP and that “Do the British still think they have some right to be treated in some "special" category?” – yes, there’s an easy argument that we DO indeed deserve special privileges.
WE saved most if not all the rest of Europe (that’s not called Germany) from occupation and degrees of genocide, the conflicts were WW1 and WW2 as you apparently haven’t heard?!
In the latter, the outcome for huge chunks of many populations would have been bleak, depending on what combination of Germany and Russia ruled afterwards.
Substantial subsets of many countries would have been fodder for concentration camps, gas chambers, medical experiments and forced prostitution, so a great many folk owe US THEIR lives – their ancestors just wouldn’t have survived.
“But America saved everyone” I hear some cries. Yes, they were a huge help, of course, BUT they didn’t arrive until after the Japanese helpfully bombed Pearl Harbour, December 1941. But for OUR resistance, the war in Europe would have been over, apart from Hitler and Stalin deciding how best to proceed. And in that event, USA would have had very tough decisions, may well not have come to liberate us all, and it would have been immensely more difficult.
“Oh, no” I hear a squawk, “it was all so long ago” . . . well, that argument doesn’t fly. Those ancestors would have been getting killed off through the forties and fifties, well before America got round to any belated rescue mission, once those ancestors were gone, they’re gone, no life at all for the family trees that followed. Too hard to follow?
I think a few helpful moves with passports – and throw in trading rules too – would be a very reasonable price for the EU to pay in return?