When I was getting into music seriously in the early/mid seventies, we used the term 'punk' (and maybe 'punk rock'; I can't quite remember) to refer to the American garage bands of the mid-sixties. I didn't know much of the music, just read about it, but I remember being a bit annoyed when the term started to be bandied around to refer to new music.
The first person I remember it being applied to was Nils Lofgren (!), around the time of 'Cry Tough', or maybe a bit before. Then it was the CBGB bands and then the UK shower. Obviously, the label didn't stick to Nils Lofgren but they really shouldn't have used the same term for the American and British scenes. In fact the label hardly works at all for the American scene, which was much less homogeneous than the British one: Television, Blondie and Ramones were linked by time and location but they were doing very different things.
And while there was cross pollination between the two countries (most of the influence travelling from west to east) I don't think they were too closely related. The Stooges were a big influence of British punk, but so was the pub rock scene and, to an extent, the DIY mentality going back to skiffle.
They were different.
And as a by-the-way, I only think of hardcore as an American thing. If it made an impact in the UK, in terms of producing bands, I was completely unaware of it.