I think that in the past zipper was the noun and zip the verb, usually as the phrasal verb zip up. If zip was used as a noun it was slang for speed or energy. But more recently I've observed zip being used as a noun referring to the fastener. If I didn't want to use zipper for some reason I'd go with zip fastener rather than just zip to avoid ambiguity.
Of course, in the USA only zip is also short for zip code, their numerical postal code.
Right. And in the US, zip (small z) is a verb and Zip (capital Z) is the postal Zip code, identifying where your mail should go:
I zipped up my jacket, put the Zip code on the envelope, and zipped over to the post office.
During the Great Depression my grandfather, who was a "jobber" — a trader in small lots in the New York City garment district — got back on his feet financially thanks to a really good deal he got on a lot of zippers. Some time after he'd bought and sold them at a significant profit, he realized they were stolen property. I never met him, but I'm told he was deeply regretful for having unknowingly participated in a crime but, beyond seeking religious redemption, couldn't find a way to rectify it.