I've found the words to 'Down in the Willow Garden,' and I don't think there's any relationship. The songs are quite different.

I think the title/first line was spliced onto an older song by somebody who had heard of 'Down by the Salley Gardens.' The phrase 'willow garden' doesn't seem natural to an American song at all. Where I live in America, willows are junk trees that are permitted to grow at water's edge, but nobody grows a garden full of them.

Their tendency to invade and clog pipes is one reason for that.