There was a point in time where Tommy Thumb was nearly equal to Mother Goose. This point was also where Tom Thumb underwent a transformation from a story that had been satirical, scatological and occasionally sexual to a story that would be told again and again for children. Not only that, but the story took on an educational nature, coupled with morals and spelling lessons.
It was not the only folktale that was toned down for a younger audience; even the Grimms sanitized and simplified the stories they retold.
There's also the fact that Tom Thumb was a byword for small, so a Tom Thumb book was simply a little book. In one early primer, The Exhibition of Tom Thumb, the character himself addresses children "who are little and good like myself," which I think cuts straight to the reason why he was used so often in this context to teach children.
For reference, the first full existing print version of Tom Thumb is from 1621.
Tom Thumb's Alphabet, by an unknown author, begins "A was an archer, who shot at a frog." It first appeared in A Little Book for Little Children, published in the very early 1700s. According to the Oxford Companion to Children's Literature, it was first printed in America in 1765 in the primer Tom Thumb's Play-Book: to teach Children their letters as soon as they can speak. I don't know when it was first ascribed the title Tom Thumb.
Tommy Thumb's Song Book (1744), by Nurse Lovechild (Mary Cooper), still exists reprints. Its sequel, Tommy Thumb’s Pretty Songbook (same year) features versions of many familiar rhymes, like London Bridge is Falling Down, but also some that might not be so familiar. Like one about bedwetting.
Tom Thumb appears briefly "with his pipe and drum."
The Travels of Tom Thumb over England and Wales (1746), by Dodsley. This story has nothing to do with Tom Thumb; the name is just used as a pseudonym. However, this was the first travel book for children, and described many different locations.
The History of England, by Thomas Thumb, Esq. (1749): another pseudonymous work.
The Monsters of Monster: a true and faithful Narrative of a most remarkable Phenomenon lately seen in this Metropolis; to the great Surprize and Terror of His Majesties good Subjects; Humbly dedicated to all the Virtuosi of New England. By Thomas Thumb, Esq. (1754). I have been unable to find further information on this work.
"The Famous Tommy Thumb's Little Story-book: Containing His Life and Surprising Adventures. To which are Added, Tommy Thumb's Fables, with Morals, and at the End, Pretty Stories, that May be Either Sung Or Told. Adorned with Many Curious Pictures" (1760).
Tom Thumb’s Playbook (1764) was a primer for children with alphabet rhymes and prayers.
The Exhibition of Tom Thumb/Tom Thumb's Exhibition (1787), produced by Isaiah Thomas. Here, Tom Thumb introduces and narrates lessons on geography, animals and morals.
Tom Thumb's Folio, or a New Penny Plaything for Little Giants (1779) contains first a pretty different version of Tom Thumb's life, and then some lessons on letters, vowels, syllables, and religious and moral lessons. One of the plot points is Tom teaching reading and writing to the inhabitants of a kingdom.
This one is also printed as The Life and Death of Tom Thumb, the Little Giant, and included Tom Thumb's Alphabet.
Tom Thumb's Picture Alphabet in Rhyme (c.1850) - "A is an angler, young but expert. B is a butcher who wears a red shirt."
The Lilliputian Magazine; or, Children's Repository (c.1773) recounts several moral tales including that of King Tom Thumb, monarch of Lilliputia, as he fights wars and marries a princess named Smilinda. His educational side is emphasized and his army includes regiments like the "Alphabetical Infantry" and the "Intrepid Sons of Syntax."
In Popular tales. Consisting of Jack the Giant-Killer, Whittington and His Cat, Tom Thumb, Robin Hood, and Beauty and the Beast (1810), instead of escaping the cow's belly with a laxative, Tom uses "comical tricks" of unknown nature - toning things down from previous versions. There is also an added moral: Tom has a troublesome life and dies young because his parents should have accepted God's will and been content instead of asking Merlin for a son.
In The New Tom Thumb ... by Margery Meanwell (1815), written by William Mackenzie, Tom Thumb (a descendant of the famous fairytale Tom Thumb) hurts animals and immediately gets what basically amounts to karma.
The entertaining story of Little Red Riding Hood ; and Tom Thumb's toy : adorned with cuts (1820?) features a version of the "Boy Who Cried Wolf" fable with a bull playing the role of the wolf. The title is somewhat baffling, as neither Tom Thumb nor toys make an appearance.
The history of little Goody Two-Shoes : to which is added, The rhyming alphabet, or, Tom Thumb's delight (1820's). "A was an angler, and he caught a fish."
Reading made quite easy and diverting: Containing symbolical cuts for the alphabet, tables of words of one, two, three and four syllables, with easy lessons from the Scriptures (c.1840), listed Tom Thumb as author, but otherwise did not mention him.
The Novel Adventures of Tom Thumb the Great, Showing how He Visited the Insect World, and Learned Much Wisdom, Etc, by Louisa Mary Barwell (1838). This is a straightforward storybook, but with plenty of moralizing and discussion of insects and animals.
Extraordinary Nursery Rhymes and Tales: New Yet Old (1876) is another one that changes up the story, focusing mainly on Tom growing up and finding a wife.
Some were straightforward retellings of the fairy tale, although with small variants or excerpts - such as this one from around the 1850s, and History of Tom Thumb by Joseph Crawhall, a poem version that I haven't encountered before.
The 19th century was filled with adaptations like Park's Tom Thumb (1836), The History of Sir Thomas Thumb, by Charlotte Mary Yonge (1855), The pretty and entertaining history of Tom Thumb (c1820), The History of Tom Thumb and Other Stories by an anonymous author, and William Raine's An Entertaining History of Tom Thumb.
It also appeared in Halliwell's Popular Rhymes and Nursery Tales (1849) and Joseph Jacobs' English Fairy Tales (1890).
Tom Thumb is a less popular character today, but his name still occasionally pops up as a byword for miniature.
This website is based on my research into folklore.