I'm from the UK (England in particular) - I think most people from England would say they are British (sometimes depending on context / person / non-requirement for brevity: 'from the UK' instead).
Perhaps a higher proportion of non-English UK people would answer otherwise.
If I read that correctly, London seems to be the only place where a majority of people see themselves primarily as "British"?
It's nearly 10 years old, but I don't think things are going to have changed to make people feel substantially more "British", certainly not here in Scotland!
Perhaps a higher proportion of non-English UK people would answer otherwise.