I do not know anything about the development or uses of majuscule letters in the Middle Ages, so I will not try to answer your question. Instead I will answer a different question: What are the capitalization conventions used for month names by publishers of Neolatin? I don't have a large and comprehensive database of Neolatin, but a small and rather biased selection based on what I can find on my own bookshelf, so this represents a personal view.
I'll start with the "authorities"; these are textbooks or dictionaries addressed to Anglophones telling us how to compose in Latin. Then I'll move on to Latin bibles; the Neolatin prefaces to these editions are often signed and dated. Then I'll cover some news sources and fiction.
Bradley's Arnold Latin Prose Composition #538 gives some examples of dates: ad Kalendas Maias, "by the first of May", in Nonas Iunias, "for the fifth of June", and Idibus Martiis, "on the 15th of March". Note that not only are the Month adjectives capitalized, but the day names (Kalendae, Ides, Nonae) are as well.
Milena Minkova's Introduction to Latin Prose Composition (p. 127) tells us to capitalize month names.
The dictionary Lewis and Short has the names of the months. If you look them up, you'll find that the names are capitalized in the example texts given in the definitions.
So much for the authorities. All the authorities I know are written for Anglophones, so their target audience may bias which convention they choose.
The Nestle-Aland Novum Testamentum Latine has two prefaces. The first preface is signed by Kurt Aland and Barbara Aland: Monasterii Westphaliae, die 30 Septembris 1983, in memoria S. Hieronymi, Patris Ecclesiae, "in Munster, Westphalia, the 30th day of September 1983, on the memorial of St. Jerome, Father of the Church." The preface to the 2nd Edition is dated die 28. Martii 1991.
In the Stuttgart Vulgate, Roger Gryson dates his signature like so: *Lovanii Novi, in festo S. Marci Evangelistae, a.d. VII. Kal. Mai., anno p. Chr. n. MDCCCCXCIV, "Louvain-la-Neuve, on the Feast of St. Mark the Evangelist, April 25,1994."
The Colunga-Turrado Biblia Vulgata has a lot of front matter, all of it dated, and the month names are invariably in lower case. There are dozens of examples, but I'll only copy a few here: die 18 novembris 1893, die 15 septembris 1920, 23 iunii 1905, 1 iulii 1933, etc., etc.
Peter Needham's translation into Latin of the comic novel Harrius Potter et Philosophi Lapis contains several examples of the fictional characters exchanging mail. In Chapter 4, we read a formal letter sent by a school of witches and wizards to a new student. The letter closes with schola aperietur Kalendis Septembribus. strigem tuam exspectamus non serius quam Pridie Kalendas Augustas. "Term begins on 1 Septembe. We await your owl no later than 31 July. (N.B. in the zany, crazy world of Harry Potter novels, witches and wizards send their letters by means of owls.)
Carolus Egger's book Omnia Dici Possunt Latine, published by the Vatican's Latinitas foundation, contains many news stories from the 1990's translated into Latin. Naturally, most of the news stories contain dates, and the month names are always capitalized. Examples: die XVIII mensis Maii, Dies XXX mensis Iunii*, etc.
So the upshot of all this is that it is a mixed bag. Some publishers capitalize month names, and others don't.