I made reference to "spoken" not "written" English to avoid a discussion in linguistics. No interest whatsoever in linguistics.
Assuming one can get tell by context what variance for similar items-intend.
The implication of your post was that spoken/written English was basically universal, but that there is effectively no carry-over from ASL to BSL - is that correct?
If so, it's not entirely correct.
While ASL and BSL are quite different (the main difference actaully being fingerspelling) that there is actaully a fair amount of signs that are the same, or similar between ASL and BSL - enough that fluent users of one can often understand at least the general context of the other quite quickly. (as an example someone posted a 2year old and their mum having a conversation in BSL - I don't know BSL, but knowing ASL, I was able to understand about 75% of what was said without using any subtitles etc to translate the BSL to text)
Part of this is because in many Signed Languages there are a number of object and action signs which are iconic, or have iconic "roots". This makes it quite easy for someone who understands how iconic signs "work" to apply that knowledge to another signed language (for example ASL, BSL etc).
On a side note, countries that share similar cultures, gestures, body language cues etc tend to share iconic signage more than to different cultures - meaning that ASL, BSL, LSF, LSQ etc share more in common (thus easier to understand) than ASL and Chinese, Israeli, Japanese, Russian sign languages.