ASL has no linguistic relationship to English, other than a few expressions that are essentially loan words; likewise, LSF (French Sign Language) has no relationship to French.
Signed languages do not know borders, nor the expanses of oral languages. ASL is spoken in the US and Canada, and also in much of northern Mexico, but it is not used in education in Mexico. ASL, or a language so close to ASL that they are mutually intelligible, is also used in a variety of countries teachers from the US (often through the Peace Corps) were instrumental in setting up Deaf ed. programs. I know the first Deaf person to be accepted into the Peace Corps, who spent almost 12 (albeit, non-consecutive) years in Deaf ed. with the Peace Corps. She was my Art History teacher at Gallaudet.
As for ASL and other signed languages being mutually intelligible, I was able to easily and comfortably have manual conversations in Costa Rica. There are two additional letters in their alphabet, which I learned (one for the n+tilda, and one for ch), and because my Spanish is pretty good, I could fingerspell. The two letters are from Spanish fingerspelling, which makes me think the signed language used in Spain is probably derived from LSF, as ASL is. This means LSE (Lenguaje de Signos Espana) is passably intelligible with LSF, and therefore with ASL as well.
I once ran into some French Deaf people in DC, and we had a conversation. It was halting, and had a few fits and starts, and also would not have been as successful had I not been able to spell French words (my French grammar is very poor, but I know lots of vocabulary). LSF and ASL separated around 1840, whereas I suspect ASL was exported to Costa Rica no earlier than the establishment of the Peace Corps.
Also, just to further my point, Mexico has its own signed language that is NOT derived from LSF nor LSE. Meanwhile, a number of countries in Southeast Asia use a variety of ASL.
That’s enough. I could write more on why signed languages are not similar to oral languages, but I don’t want this post to be any longer.