why don't British people say: "Everybody mustn't take drugs".
First, who says it's grammatical? Second, who says only British speakers don't use it? And, third, it's neither grammatical nor colloquial; it's a spin, or play on Bob Dylan's song Everybody must get stoned.
The pronoun everybody is indefinite. It refers to a broad group of people, a group that has little specification, and so it doesn't work with modal mustn't, because in the negative that modal requires a subject that refers to a specific person. Consider this. The pronoun somebody, like everybody and anybody, is indefinite, but if we make it definite, that is, specify the person it refers to, then it works with negative must. Like this,
Specific: Somebody I know mustn't take drugs.
Non-specific: Somebody mustn't take drugs.
Does that help?