According to some sparse notes I made about 30 years ago, based on the collection of Thomas Hudson's little songbooks at the University of Wisconsin, there were 13 books (of about a dozen songs each) of the which the last was undated, and the 12th was of 1831.

At any rate "Betsy Baker" is in his 'Comic Songs by Thomas Hudson: Fifth Collection', 1824. My notes don't indicate whether he cited a tune for it. He usually didn't, but in the same collection is "Jack Robinson" to the tune of "The College Hornpipe".

In the 13th book is a reworked version of "The Cruel Mother" which Hudson called "The Country Tragedy" to be sung to "Wellady".

There are a 3 or 4 of his early songbooks in the Library of Congress, but I haven't yet relocated my notes on them. Hudson was a greengrocer by trade and published and sold his songbooks from his grocery shop in London.