How did the word charlatan find its way into English?
from French charlatan "mountebank, babbler" (16c.), from Italian ciarlatano
The earliest entry in the OED for 'charlatan' is:
[1607 B. Jonson Volpone ii. ii. sig. D4v The rable of these ground Ciarlitani, that spread their clokes on the pauement.
and by 1618 the spelling is the modern one:
1618 D. Belchier Hans Beer-pot sig. Djv I think the Serieant is grown Mountebancke To cling by shifts, hey, passe, passe, Italian grown; a sharking Charlatan.
(if you have the ability to see the full entry, I recommend it. The definitions for 'charlatan' read like an extended Johnson critique, using very colorful terms much less common).
Not knowing the personal educations of those mentioned, this leads me to believe that Jonson was acquainted with Italian and Belchier French and that they or whoever they knew were just using the word as a learned borrowing.
I agree with Mitch's answer but want to elaborate on the prior Spanish, French, and Italian importations of the word into English. French remains the most likely point of origin, but Spanish and Italian uses of the word would have been present in late 16th century and early 17th century books.
The argument for the word being primarily from French is strengthened by earlier evidence. The earliest non-lexicon entry I can find is from a 1590 book whose full title bears witness to its translation from French: The coppie of the Anti-Spaniard made at Paris by a French man, a Catholique. Wherein is directly proued how the Spanish King is the onely cause of all the troubles in France. Translated out of French into English. In a section that protests a Spanish-appointed rector of the University of Paris, who advocated for an inquisition to deal with Protestant reformers, the author writes a series of questions pointing at the rector's hypocrisy:
Why is [Inquisition] not vsed in the low Coun∣tryes? how comes it to passe that hee who hath such excel∣lent and soueraigne remedies against this disease can not cure himself? How can he promise helth vnto vs, whē he him selfe is full of sores and vlcers? Wherefore should we rather thinke him a skilfull Phisiition, then a pratling Ciarlatan?
A prattling charlatan! The text's claim that it is a translation suggests that the author may be carrying a particularly-appropriate loan-word from French.
Cotgrave's 1611 Dictionary of the French and English Tongues also features the word, and is the lexicon that the OED refers to. The lexicon presents French headwords with English synonyms. Cotgrave's entry reads:
A Mountebanke, a cousening drug-seller, a pratling quack-saluer, a tatler, babler, foolish prater, or commender of trifles.
It is accompanied by other headwords charlatanerie and the abbreviated forms charlater and charlaterie.
Meanwhile, Spanish provides the earliest possible mention of charlatan. In a Spanish-English lexicon from 1590, The Spanish Grammar by Antonio del Corro, the entry charlatan is defined as the following:
a prattler, a chatter.
In Spanish this has more of a sense of being a chatterbox, in line with the French sense of "babbler," or the method by which a charlatan deceives people. Early English Books Online also gives a result for the entry being in the 1599 edition of A Dictionarie in Spanish and English, though I don't have full-text access to it. So Spanish influence could have influenced the word, though I have no non-lexical sources to support that. The rest is guesswork - English authors were reading Spanish texts, but it'd take a lot of work to find or rule out a Spanish intertext.
Finally, the direct influence of Italian remains possible. In a 1598 Italian-English lexicon by John Florio, A vvorlde of wordes, or Most copious, and exact dictionarie in Italian and English, this series of headwords appears:
Ciarla, Ciarleria, a chat, a prattle, a tit∣tle tattle, or scoulding.
Ciarlare, as Chiacchiarare, as Cian∣ciare.
Ciarlatore, Ciarlatano, as Ciaratano.
Ciarliere, as Ciaratano, or Ciancione.
Ciarlone, as Ciancione.
Jonson's Volpone (1607), set in Italy, could have been drawing off of Italian when he described Ciarlitani:
1607 B. Jonson Volpone ii. ii. sig. D4v The rable of these ground Ciarlitani, that spread their clokes on the pauement.
In short, the French influence of the word is unquestionable, but influence from Italian and Spanish seems plausible as well. It's a continental word that spoke to a common perception of unscrupulous prattlers and sellers.