"I learned that Nyth a Gwcw from two Welsh musicians, but they called it Nyth a Gog."

I think that may be a north Welsh/ south Welsh difference.

The tune is found all over the place with both titles (and variations thereof) There are songs with words based on both titles too.
Versions I know of include:
Numerous Cotswold Morris Tunes
The Welsh one
"Jackie Tar"
"Come Ashore Jolly Tar, with your Trousers On" *
"The Cuckoos Nest" from Laurence Leadley, the fiddler of Helperby in north Yorkshire.

This is The Leadley Cuckoo's Nest which is almost note for note a major key version of the usual minor key session version of Jacky Tar.

* believed to be a reference to sailors keeping a clean pair of trousers on board for shore leave.