The cards as woodcuts

The Rothschild cards were printed first and then painted,  according to Gerardo Ortalli in "The Prince and the Playing Cards Ludica, annali di storia e civiltà del gioco 2, 1996, p. 194 (I thank as "Huck" on Tarot History Forum for this reference, given at http://forum.tarothistory.com/viewtopic.php?f=11&t=1005&start=10#p14986). There are clearly lines on the cards in addition to the painted areas, visible for example on the Knight of Swords, here; but are these from woodcuts?  Fiorini disagrees, in footnote 3, p. 54:

Ho potuto verificare personalmente la tecnica esecutiva delle carte, che non presentano traccia alcuna di incisione, grazie alla cortesia di Pascal Torres, conservatore presso il Cabinet Rothschild. Ringrazio per i preziosi consigli anche Alberto Milano e Thierty Depaulis che, dopo un recente sopralluogo presso la collezione del Lortvre, ha gentilmente confermato la mia ipotesi.

[3. I was able personally to verify the techniques of the cards, which do not show any trace of incision, through the courtesy of Pascal Torres, curator of the Cabinet Rothschild. Thanks to Alberto Milano for valuable advice and to Thierry Depaulis, who, after a recent inspection of the collection at the Louvre, has kindly confirmed my hypothesis.]  . 

I do not know if that settles the issue or not. Depaulis is probably the leading authority on such cards currently, which surely counts for something. But when did woodcuts first appear in Florence?

Ortalli does not give any dates other than before 1476. But when Bernardino preached against cards in 1424 Bologna, he probably had in mind cheap ones, because that's what his audience would have used.

According to Arthur Hind, An introduction to the history of woodcut, 1935, p. 96, the first extant dated woodcut is a Madonna of 1418 Flanders, followed by a St. Christopher of 1423 Augsburg. These are sacred images, worthy of being preserved; cards weren't. Given the popularity of cards, and the conservatism of monks, I would guess that cards were earlier than these sacred productions. About playing cards from woodcuts, Hind says (p. 84):

In the first place the production of playing-cards must have been a thriving industry, especially at Ulm (2), at the end of the XIV and beginning of the XV century. In spite of the fact that no existing pack of cards can be dated with any certainly before about 1450, this does not rule out the probability of woodcuts having been used by about 1400, if not earlier, in their production.
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2. See Felix Fabri, Tractatus de Civitate Ulmensi, herausgegeben von G. Veesenmeyer, Litterarisches Verein in Stuttgart 1889, pp. 145, 146.

Hind adds (p. 84) that

Through the very nature of their use cards are the most unlikely things to be preserved.

Prints on paper can also be dated by their style. Hind observes (p. 79):

It seems unlikely that any large supplies of paper were available before the latter part of the XIV century 2, and this was probably an important factor in determining the period at which the printing of pictures was introduced. Judged from the style of their design, the woodcuts which appear to the be the earliest printed on paper should be dated about 1400, hardly later, and hardly more than ten or twenty years earlier.

On woodcut-produced cards in Italy, there is also documentation. Here is Hind p. 80:

Kristeller notes a certain Federico de Germania who sold cartas figuratas et pictas ad imagines et figures sanctorum at Bologna in 1395, and a card-maker (pittor di naibi) of Florence, Antonio di Giovanni di Ser Francesco, who declares among his property in 1430 'wood blocks for playing-cards and saints'. 3.
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Kupferstich und Holzschnitt, 1922, pp. 20 and 21. There is no proof that Federico de Germania printed his cards from blocks, but as the document in which he is mentioned is an action against him for coining false money, Kristeller thinks it highly probable that a die-cutter would have been equally conversant with block-cutting. It is worth noting in this relation a fact emphasized by Bouchot (Ancetre de la gravure su bois/i], p. 22) that in France early woodcuts seem to have sometimes been regarded as [[i]malfacons (contraband), a sort of false coin in the eyes of the build of Imagiers.

Hind goes on to note, of course, St. Bernardino's diatribe in 1424 to persuade the players to burn their cards. We have no way of knowing, to be sure, how common these cards were at the time.
 Another woodcut is dated to 1428 (Grove Encyclopedia of Medieval Art and Architecture, Volume 2, by Colum Hourihane, p. 409, in Google Books); it is a "Virgin of the Fire", 1428, revered because it was saved from a fire, Hourihane says. It is not in the International Gothic sylte, so its authenticity has been doubted; Hourihane, however, sees affinities with the facial features of early Venetian art and of a woodcut of The Trinity with Saints.

One way in which prints on paper from woodblocks might have been produced early on is from the woodblocks used for making textile prints. We know about them from the numerous Late Gothic depictions of sumptuously decorated clothing, including that depicted in the painted part of the Rothschild cards. Hind observes that textile printing was common throughout the middle ages, although not with pictured figures until the second half of the 14th century (p. 67). He gives numerous examples on textiles from about 1400. He adds (p. 69):

The printers of textiles may also have sometimes pulled impressions on paper as patterns, and some of the earliest prints on paper (known for the most part only in single impressions) may have been made for this purpose.

He gives several examples; the only problem is that there is no extant corresponding textile, so this use is only "probable". I would imagine that such a "probable" use would apply in Florence, which it is my impression was even in the early 15th century a center of textile production..