Why do British people never sound British when they sing?

You need to listen to Bellowhead, Blur, False Lights, John Martyn, Lau etc plenty do

well I saw a documentary about the early days of pink Floyd that said something interesting regarding this question. they said during the "British invasion" all the British bands were trying to sound American (like the rolling stones) and finding inspiration in American black culture like blues and jazz but one exception in this was Sid Barret, first singer of pink Floyd, who actually sounded British when he sang and this was one of the things that made him stand out.

Because there is no ‘British’ sound
We have many sounds, we’re like an orchestra

Tom Jones is Welsh and sounds Welsh :)

The same reason Americans don't sound American when they sing, I'm English and sound French when I sing....

The subtleties of accent are often lost in the stretching of vowels.

Maybe it's because they almost always sing in the same language they normally speak.

Abba sang their songs in English (and Spanish on some versions of their albums), and while I couldn't tell that they were Swedish just from listening to them sing, I could tell that English wasn't their native language, because a bit of their native Swedish accent slipped through.

Music has its own rhythm and accentuations which soften accents.

Songs follow a specific vocal pattern, they don't rely on accents. Having said that, listen to Ed Sheeran's perfect. When he says "your heart is all I own" , "all" sounds quite different to me. There's an influence of their accents, but you wouldn't notice it unless you pay attention to the details without enjoying the song.

they oftentimes do, actually.
[youtube=https://www.youtube.com/watch?v=lv8k0VI9tBc]

I've wondered that myself--- unless they're Australian--- then they sound british

The same goes for Oriental and Asian people

When they sing in English at least

I have no clue to why that is

No such thing as british sound