I'm not aware of historic English alphabets that include the dollar or pound sign or any sort of punctuation. On the other hand there is a lot of evidence of & being part of the alphabet. Wikipedia has or links to plenty of cases:
* Old Saxon alphabet: https://en.wikipedia.org/wiki/Old_English_Latin_alphabet#/me...
* The Dixie Primer, for the Little Folks: https://docsouth.unc.edu/imls/moore/moore.html#moore5
* Byrhtferð's Old English Alphabet: https://www.evertype.com/standards/wynnyogh/thorn.html
* My Own Primer, or First Lessons in Spelling and Reading: https://shadycharacters.co.uk/2011/07/the-ampersand-part-2%C...
None of these are songs.
I’m not at all convinced that it was ever what you might call a letter of the alphabet. Although it seems to not have been rare to group it with the letters in some way, I have received the impression it wasn’t particularly common, and that it wasn’t how people generally thought of it. Nowadays we would draw a clear distinction between letters, numbers and symbols (maybe we have more symbols? though certainly fewer ligatures/abbreviations like Ƿᵉ), and call the alphabet just the letters, but I get the vibe that maybe “alphabet” wasn’t so strictly just the letters. But I handwave liberally and provide no sources in this mostly-uneducated suspicion of mine.