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This is covered by Bonnie Blackburn and LeoFranc Holford-Strevens in their book The Oxford Companion to the Year: An exploration of calendar customs and time-reckoning (1999, reprinted with corrections 2003, p. 671). They call the method we use today the "forward count". The forward count begins to be attested in the fifth century, and makes greater inroads from the eleventh century. But the Roman style is not considered less Christian; Gregory XIII used the Roman date style in his bull ordering the reform of the calendar in 1583.
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This article on the Codex-Calendar of 354 may be of help. This and some encyclopedia articles on kalends, nones and ides implies that the Roman form of the calendar was tied to the old pagan religions, and that it competed with the 7 day weekly cycles of the church calendar.
So though the words remain in the language, they drop out of the calendar with the collapse of the pagan aristocracy.