Why do many people from the UK dislike the term ‘British English’ just because they have their own regional dialect or accent?

You’re conflating accent with language.

British English is a legitimate term that refers to the use of English in Britain as opposed to the English used in, say, the U.S. or Australia. But that’s a language issue, not accent.

The term British accent is what causes Brits to roll their eyes, because it usually refers to what is called Received Pronunciation, and there are dozens other British accents.

What Brits really dislike is when software refer to British English as ‘British English’, and American English as ‘English’—as though English had been invented in America, and British English was just a derivative. Case in point:

 

(Originally written in reply to a question at Quora.com).