How do I go about making this happen?
I'm a British user, and I don't see the point. "Greatest common divisor" isn't an opaque term, and it is the one which I've encountered and used since, at the latest, partway through secondary school. If you're concerned about tag intelligibility for people who don't follow an American curriculum, the place to start would IMO be algebra-precalculus.
I support the synonym regardless of whether the term "highest common factor" is in widespread use somewhere or not. It is a phrase that (1) clearly cannot mean anything except greatest common divisor, and (2) is sometimes used. We have one question currently bearing that tag, so certainly someone used it. If no one ever uses it, it's still harmless.
On Wikipedia, it is considered good practice when creating a new article to create redirect pages from commonplace alternative names, alternative capitalizations, alternative spellings, probable misspellings (thus "monestary" redirects to "monastery", etc.), commonplace misnomers, etc. The practice is at worst harmless.