There are two parts to this answer.

First, remember that Joan was tried, convicted, and burned for heresy. When we say she was tried by "the English," it's true in a sense that the participants were either English themselves or supporters of England, and the trial took place in an English stronghold. But she was tried in a Church court led by a bishop, Pierre Cauchon (who supported the English on the premise that they were not the particular French king he hated). Her trial was essentially political, but it was still a religious crime and an ecclesiastical court. From that standpoint, "the Church" was not picking sides in the 100YW. (For assorted reasons, we have no converse English prophet as a test case for the other side.)

But that's just a bit of pedantry that pushes off the question, why would it be credible that God pick sides when the enemies are of the same religion and not fighting over religion. For that, we need to back up chronologically a bit.

In the later Middle Ages, the Church and especially the papacy contended for power in the "secular" realm as well as striving for a spiritual monopoly. Political factions from Roman families to kings of France saw the papal throne as something to be controlled for their own earthly benefit, and the popes themselves were hardly immune to being embroiled in noble politics (since they, along with basically the entire Church hierarchy with any semblance of power, came from the elite or at least the moneyed).

One of the more...special...ways this secular/spiritual fusion caused conflict was when a king or emperor would take it upon themselves to appoint "the true pope" when they did not appreciate the cardinals' choice (or "cardinals'" choice).

With the twelfth century schism, we see the rise of prophets speaking divine revelation about the mess. Hildegard of Bingen-as-prophet isn't technically taking sides. She claims to relay the message that God condemns BOTH sides for the disorder it causes in the Church. When asked directly about it via letter, she responded, "It does not please God that I should pronounce myself on the schism of the Church." Elisabeth of Schönau is not so reticent. As she dictates it, the angel who reveals God's messages to her says, "And you should know that the one who has been chosen by Caesar [Holy Roman Emperor] is more acceptable to Me."

From Hildegard and Elisabeth we see it is credible, to medieval people, that God take sides in an intra-Christian political conflict when Church unity and apostolic succession is at stake--God surely wants His Church to be unified.

Fast forward to the late 14/early 15C, and the Great Western Schism tears apart Europe over the Avignon versus Rome popes, antipopes, everyone gets a pope! The amazing, unprecedented development we see in the Great Schism is the rise of peasant/urban, uneducated prophets who have VERY sharpened, God-backed political opinions about the split in the Church. There are Great Schism prophets on both sides!

By this point, it was clearly credible to people at all levels of society that God could and did takes sides in political controversies. Yes, the Church was involved, but the mess transcended the spiritual realm. In the Iberian peninsula, supporting different popes became a secondary way that warring factions distinguished themselves, not a reason they disagreed.

Joan of Arc rose to prominence on the basis of her revelations in the wake of the Great Schism and its prophetic awakening. A more politically aware western Europe was equally ready to hear the words of God taking sides in a war among Christians and to condemn those words as the devil's.

Why God would take sides has not been ignored even from the start. Her judges at the initial trial pressed Joan on this, asking, for instance, in what language he her Voices spoke to her, English or French, and did he desire the deaths of English soldiers. Mary Gordon, in her brief, lively bio of Joan (2000), points out that the devil’s advocates during the early 20th-century investigation of her sainthood (i.e., those scholars who took the negative case arguing against her sanctity) likewise fretted about this. One of them wondered

”since France has been such a source of poisonous ideas and so much trouble to the Church, maybe her cause wasn’t a good one. Wouldn’t the Church, he suggests, have been better off if France had ceased to exist?”

Siobhan Nash-Marshall goes to the opposite extreme, seeing virtue in divine partisanship in Joan of Arc: A Spiritual Biography (1999). Joan, she says, “proclaimed that the larger unity [of a France united against the English] was a sacred thing—that it was a response to a divine calling” and “Joan’s claim that God wanted France to be a nation implies that nationhood itself can be sacred.”

This is too far out on a limb for me, but a few things cross my mind. First, medieval “just war” theory would sanction fighting against an invader, which the English were, no matter the (somewhat) legitimate claim they had to be considered kings of France. In this regard, from a medieval viewpoint God could “take sides” even when Christians fought. Second, Joan’s nullification trial in 1455-56 didn’t directly address the political-military questions that had been in dispute. Instead, it rehabilitated her by declaring that her first trial had been illegal and irregular, and therefore invalid (in the court’s words: “we break [the earlier judgments], annihilate them, annul them, and declare them void of effect”). Third, with the 100YW now over as of 1453, it was important to Charles VII that Joan be vindicated in order to quell any misgivings that he had been brought to the throne by a heretic (and maybe a witch); it was undoubtedly this need that expedited the nullification trial which had been stalled. Fourth, French politics in the late 19th-century helped propel Joan to sainthood. For one thing, a sainted “daughter of France” could help sustain the rekindled Catholic church in France that had been so badly damaged by the French Revolution and modernism. Finally, by the time of Joan’s canonization in 1920, the papacy had no cause to much fear treading on the feelings of the English by making Joan a saint and tacitly putting God on the side of the French. The English church had long been broken from the papacy with Henry VIII’s Reformation and its aftermath.

Pius X’s language in the document of Joan’s 1909 beatification (pre-sainthood) is not subtle about whom God rooted for. The “war with the English, pointed to the widespread and swift destruction of France.” France’s nobles “had already yielded to the English invasion” [my emphasis]. “In that noble battle [at Orleans], not without the present assistance of God, the enemy forces were driven back and overthrown by fear, and this victorious maiden.” They were “the invading English.” Her victories inspired people with “love for their country and restored piety toward God gave added strength in the face of egregious crimes [my emphasis].” She led Charles to his coronation at Reims where “Clovis I king of the Franks was washed in the purifying waters by the holy Remigius and laid the foundation of the nation of France. Thus were the enemies of the name of France [i.e., the English] fought against by Heaven. Thus, having saved her country by divine aid, the maiden of Arc fulfilled her mission.” He widens his perspective, aiming at nationalism and secularism:

For in the present time, in which the Catholic world sees and is saddened by so many great evils and in which so many who hate the name of Christian feign love for their homeland because of the ruin both of state and religion, it pleases Us to celebrate the glorious example of that most courageous Maiden.

Benedict XV’s official pronouncement of her canonization in 1920 tones down the sniff of anti-English sentiment in Pius X’s proclamation, perhaps because WWI had intervened, where France and England were allies. Here, the focus in on Joan’s “unjust condemnation” by an “unlawful tribune,” no mention of the English.

Sources: Translations of the decree of the nullification trial in 1456, the beatification in 1909, and the canonization are online here. Mary Gordon, Joan of Arc (2000), Siobhan Nash-Marshall,Joan of Arc: A Spiritual Biography (1999). Donald Spoto’s pretty wacko bio Joan of Arc: The Mysterious Life of the Heretic who became a Saint takes Nash-Marshall’s idea and runs with it. If you really believe Joan’s voices came from God, so does he! If I recall, Kathyrn Harrison’s 2014 bio, Joan of Arc: A Life Transfigured talks a little about this but I can’t find my copy.

Edit: corrected url

Edit: Not God but her Voices, his saintly messengers.