American Sign Language is rooted in French Sign Language, because a teacher at a long-running school for the deaf there came to America to teach.
British Sign Language developed in a different direction, where fingerspelling the alphabet is very different from how it is done in ASL. So even though US and England share a spoken language, signed expression developed differently.
Same deal with signed languages around the world. They generally develop locally and incorporate influences from elsewhere, much as spoken languages do, with the local dominant spoken language having the most influence.
British Sign Language developed in a different direction, where fingerspelling the alphabet is very different from how it is done in ASL. So even though US and England share a spoken language, signed expression developed differently.
Same deal with signed languages around the world. They generally develop locally and incorporate influences from elsewhere, much as spoken languages do, with the local dominant spoken language having the most influence.