As it has been observed, Tolkien does not directly say anything that would connect it within Middle Earth. However, there are several clues that, if I may be permitted, I would connect to present a theory:
The 14th of the New Year
First if all, the 14th of the first month in the Jewish calendar is Passover. It would land roughly around early April give or take a couple weeks since it is based on a semi lunar system.
Christianity believes Jesus to have died on Passover, thus securing our salvation as the Messiah.
Christian Passover Dates
Early Christians continued to observe Passover but as remembrance for the entire Passion weekend (both Jesus death and resurrection) but as the first two centuries of the church went on, churches celebrated it increasingly differently. Different times, days, oh the Jewish Passover, on a different day using different calcification. The whole ordeal became a bit of a mess until the ecumenical Council of Nicea declared the rules for Easter. There was still some disagreement but that's all that's relevant.
No longer was "Easter" held on the same day or even by the same calendar system as the old Jewish Passover, instead:
In 325CE the Council of Nicaea established that Easter would be held on the first Sunday after the first full moon occurring on or after the vernal equinox. 1
Interestingly, the result is a wide range of possible dates, but the average, by median, is April 7.
Not only that, but in Catholic tradition, the date for the first Good Friday, the day of Jesus's crucifixion, is March 25. It is true this is primarily the date of the Annunciation, but it is also connected in legend to his : Did the Annunciation and Good Friday coincide?
Though Tolkien did not like obvious allegory, he frequently uses themes and symbols that are analogous to real world ideas, concepts and history. The Messianic theme in The Lord of the Rings is impossible to miss. The title Return of the King itself hints at this as well as the recurring prophetic symbolisms such as the reforging of the sword and the details around the White Tree (sadly lost in the movies). So though Tolkien may not have intentionally or even consciously included them, allegory does exist, by Tolkien's own admittance.
So while Gondor was saved on the 14th of the new year, that they later historically shifted from 14th of first month to a fixed date in their own calendar would be nearly perfectly analogous to Christianity's fixing of Easter's date apart from the old Jewish calendar. It being placed on the 14th of the new year AND March 25th is simply too great a coincidence to ignore for an educated Christian mind (which Tolkien certainly had).
Further more, Tolkien also marks this as the beginning of a new Fourth Age of Middle Earth, much like the Christian calendars understood post Christ history as a new era.
I am not the first to make this connection to the Catholic day of Christ's death, either:
J.R.R. Tolkien: Author of the Century,
By Tom Shippey
Tolkien's reasons for this cannot be known for sure, if they were even known to him, but it certainly reinforces the Messianic theme of salvation and the defeat of evil and likely was a small and perhaps private homage to his Catholic faith.
If I was forced to guess, I would say it started unconsciously or coincidentally, but, upon realization, was later developed within his own calendar system to allow the event to synchronize in Middle Earth calendars and to parallel our Earth's calendars as the precision of it all defies coincidental or subconscious means.
When writers were asked about intentional symbolism in their writing, Ray Bradbury had this to say:
“No, I never consciously place symbolism in my writing. That would be a self-conscious exercise and self-consciousness is defeating to any creative act. Better to let the subconscious do the work for you, and get out of the way. The best symbolism is always unsuspected and natural."
And Isaac Asimov replied:
Isaac Asimov: “Consciously? Heavens, no! Unconsciously? How can one avoid it?”