According to the OED, *wyrm* (in the spelling *worm*) was first used to refer to the earthworm  and similar creatures in Middle English, and even in early use it seemed to refer to a wide range of creepy crawlies. The [Middle English Dictionary][1] documents all of these meanings. 

*Dragon* also emerged in Middle English as a loan word from French, transmitted through some combination of bestiaries, saints' lives like [St. Margaret's][2], Biblical translations, and romances. I would consult the [Middle English Dictionary][3] quotations to see the sheer range of sources *dragon* appeared in. 

Throughout Middle English the two terms coexisted, both referring to dragons. In the early modern period, the more modern meaning of *worm* has already taken over its earlier large and serpentine sense. The [Lexicons of Early Modern English database][4] shows the dominance of *worm* referring to insects and *small* serpents even in early entries. In Latin-English dictionaries they are often *vermis*, Latin for the modern worm. By Samuel Johnson's Dictionary (1755) he defines "[worm][5]" primarily in this way: 

>  1. A small harmless serpent that lives in the earth.

>  2. A poisonous serpent.

Then parasites, then silkworms, then grubs, and then more figurative senses. Wyrm is absent. In contrast, *dragon* even in the 16th century was the primary translation of draconic words. *[The Dictionary of Sir Thomas Elyot][6]* (1538) glosses *draco*, *-onis* as "a dragon." Such usage continues down to Johnson's *Dictionary*, where "[dragon][7]" was defined in its modern meaning: 

> 1. A kind of winged serpent, perhaps imaginary, much celebrated in the romances of the middle age.

So save in specialized or archaic uses, *dragon* outpaces the draconic sense of *worm* by the 16th century. 

---

As someone who has studied medieval literature for a long time, my sense is that *worm* was fast diminishing in Middle English save in texts that hearkened back to Old English usage like the alliterative *Sir Gawain and the Green Knight* (late 14th c.), whereas *dragon* became the preferred expression in popular saints' lives and romances. I consulted a few popular sources and one database to work through this hypothesis: 

(1) Sir Thomas Malory's ***Le Morte D'Arthur*** (written ~1470, printed 1485), the Arthurian romance seminal to 19th and 20th century adaptations of King Arthur and his knights, features the *dragon* at several key points, including in a dream King Arthur has before fighting a giant. I'm writing from the Winchester MS and updating the spelling but not the syntax: 

> As the King was in his cog and lay in his cabin, he fell in a slumbering and dreamed how a dreadful **dragon** did drench much of his people, and come flying on wing 
 out of the west parts. ...

*Dragon* occurs several times throughout the story. *[Worm][8]* and *[wyrm][9]* never appear. A later epic romance, Edmund Spenser's *Faerie Queene* (1590, 1596) fits the same pattern: *[dragon][10]* appears 13 times and *[worm][11]* only appears as "gealouse worme" and "worm-eaten." That's remarkable, since the Redcrosse Knight defeats a dragon at the end of book 1, so there was a perfect opportunity to fight a *worm*, but Spenser never chose that word for that purpose. 

(2) The ***Wycliffe Bible*** are an early complete translation of the Bible completed by religious reformers under the direction of John Wycliffe. **[Dragon][12]** appears 47 times and usually refers to serpents or monsters:

> (Revelation 12) : And another sign was seen in heaven; and lo! a great red dragon, that had seven heads [having seven heads], and ten horns, and in the heads of him seven diadems.

**Worm** appears 33 times, usually referring to small creatures: 

> (Isaiah 51:8) : For why a worm shall eat them so as a cloth, and a moth shall devour them so as wool; but mine health shall be without end, and my rightfulness into generations of generations. 

So even around 1400 a series of translators were distinguishing worms and dragons. 

(3) An [Early English Books Online search for both worm/wyrm and dragon][13] yields over 1500 results. I've looked at the first several, and so far the words appear to not overlap in meaning. Now, that doesn't preclude *worm* referring to large serpents in some specialized or literary uses, but even the OED entries for Shakespeare and Milton refer to small serpents. I can find no association with anything dragon-like in English after 1500, suggesting that the association after that point is *rare* or intentionally *archaic*: 

> a1616   Shakespeare Antony & Cleopatra (1623) v. ii. 238   Hast thou the pretty worme of Nylus there, That killes and paines not?  

>1667   Milton Paradise Lost ix. 1068   O Eve, in evil hour thou didst give eare To that false Worm .   


  [1]: https://quod.lib.umich.edu/m/middle-english-dictionary/dictionary/MED53443/track?counter=1&search_id=799654
  [2]: https://en.wikipedia.org/wiki/Margaret_the_Virgin
  [3]: https://quod.lib.umich.edu/m/middle-english-dictionary/dictionary/MED12521/track?counter=1&search_id=799654
  [4]: https://leme.library.utoronto.ca/search/quick
  [5]: https://leme.library.utoronto.ca/lexicon/entry/1345/42475
  [6]: https://leme.library.utoronto.ca/lexicon/entry/63/9980
  [7]: https://leme.library.utoronto.ca/lexicon/entry/1345/12141
  [8]: https://books.google.com/books?id=DYcaAQAAMAAJ&dq=concordance%20to%20Malory&focus=searchwithinvolume&q=worm
  [9]: https://books.google.com/books?id=DYcaAQAAMAAJ&dq=concordance%20to%20Malory&focus=searchwithinvolume&q=wyrm
  [10]: https://books.google.com/books?id=iWTepQNJYVcC&dq=faerie%20queene%20concordance&focus=searchwithinvolume&q=dragon
  [11]: https://books.google.com/books?id=iWTepQNJYVcC&dq=faerie%20queene%20concordance&focus=searchwithinvolume&q=worm
  [12]: https://www.biblegateway.com/quicksearch/?quicksearch=dragon&qs_version=WYC
  [13]: https://quod.lib.umich.edu/cgi/t/text/text-idx?ALLSELECTED=1;xc=1;g=eebogroup;type=boolean;rgn=works;q1=worm;op2=or;q2=wyrm;op3=and;q3=dragon;singlegenre=All;c=eebo;c=eebo2;view=reslist;subview=short;sort=occur;start=1;size=25