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).
This happens all the time.
Just go to the local British High Commission, fill out the relevant form, and pay a fee (around £123 when I last checked). They’ll send you a new British accent in the post within a week (assuming, that is, that you are British).
The Indian accent sounds a little like the Welsh accent—but all regional accents in Britain are made fun of, in an affectionate way.
I can recount three stories about this—so you can judge for yourself.
(My answer to this question at Quora.com)
There are two issues here: speaking in a reasonably native accent in more than one language, and speaking with more than one native accent in the same language (e.g., various regional accents in England, Scotland, Ireland, the U.S., Canada, etc. when speaking English).
It is common courtesy to try and adopt the standard pronunciation of a country when you learn its language—if only so as to make yourself more intelligible to the locals and to put them more at ease. But switching between different native accents in a given language requires training, and is something that you would do only if you were a professional actor or standup comedian and needed it for your work.