Anecdotally, if you ask anybody what a druid is, or what a druid does, many will say they are mythological shapeshifters that are tuned with nature.
Druids were, in fact, political advisors and figures of authority for Celts around the time of the Roman empire and beforehand.
I thought relationship came from the Church trying to label Druids as beasts, trying to downplay their role and religion. This is only partially accurate. I've found that the Romans, and the church more specifically, would label druids as magicians and practitioners of magic. However, there is no account of druids turning into animals.
When in history did it become widely accepted that Druids could shapeshift into animals?
Edit: My Research
As stated by user sempaiscuba♦, Laignech Fáelad is the only referenced shapeshifter in the book. The problem is that Laignech Fáelad is considered a legendary warrior. It is reasonable to assume that Laignech Fáelad would have not been a druid in this case because druids of the celts tended to be advisors and scholars. They would be involved in political discussions and perform sacrifices. It is important to note that he was a descendant of a tribe of werewolves that were related to the kings of Ossory. This tribe was present in eastern Ireland which would place them where the Celts would practice. It is reasonable to assume that this tribe would include druids and shamans. However, it is not a definitive answer that druids were shapeshifters.
There is a reference on https://rpg.stackexchange.com/questions/112244/is-the-shapeshifting-druid-an-original-dd-invention asking if D&D invented druids transforming. The answer had priests originally as the shapeshifters. It was stated that, "Gary Gygax himself has stated that the druid was based on Caesar's description of druids in Commentarii de Bello Gallico (at least, according to James Maliszewski of Grognardia). That description did not involve shapeshifting, but it does draw the connection between D&D druids and Gaulish priests of that era." It is incorrect that Julius Caesar called the Celtish druids shapeshifters. A more accurate translation can be found at http://www.eyewitnesstohistory.com/druids.htm and says
"The Druids are in charge of all religious matters, superintending public and private sacrifices, and explaining superstitions. A large crowd of young men, who flock to them for schooling, hold the Druids in great respect. For they have opinions to give on almost all disputes involving tribes or individuals, and if any crime is committed, any murder done, or if there is contention about a will or the boundaries of some property, they are the people who investigate the matter and establish rewards and punishments. Any individual or community that refuses to abide by their decision is excluded from the sacrifices, which is held to be the most serious punishment possible. Those thus excommunicated are viewed as impious criminals, they are deserted by their friends and no one will visit them or talk to them to avoid the risk of contagion from them. They are deprived of all rights in court, and they forfeit all claim to honors.
There is one arch-druid of supreme power. On his death, he is succeeded either by someone outstanding among his fellows, or, if there are several of equal caliber, the decision is reached by a vote of all the Druids, and the election is sometimes managed by force. At a fixed time of year they assemble at a holy place in the territory of the Carnutes, which is thought to be the center of Gaul. Anyone with a grievance attends and obeys the decisions and judgments which the Druids give. The general view is that this religion originated in Britain and was imported into Gaul, which means that any keen student of Druidism now goes to Britain for information. . .
The whole Gallic nation is virtually a prey to superstition, and this makes the serious invalids or those engaged in battle or dangerous exploits sacrifice men instead of animals. They even vow to immolate themselves, using the Druids as their ministers for this purpose. They feel that the spirit of the gods cannot be appeased unless a man's life is given for a life. Public sacrifices of the same sort are common. Another practice is to make images of enormous size, with the limbs woven from osiers [willows]. Living human beings are fitted into these, and, when they are set on fire, the men are engulfed in the flames and perish. The general feeling is that the immortal gods are better pleased with the sacrifice of those caught in theft, robbery or some other crime. But if a supply of such criminals is lacking, then they resort to the sacrifice of completely innocent victims. . . "
TL;DR: Julius Caesar was calling the Celtish people superstitious, and detailing their rituals.
There is a source http://druidsegg.reformed-druids.org/newslughnasadh10-16.htm that details Celtish myths of turning people into animals for various reasons. The source itself is by no means scholarly, but it does include scholarly references that do corroborate what the article is saying.
Indeed, there are many myths of the Celts that suggests that people are turned into animals for a variety of reasons.
A Catholic Encyclopedia, suggests that the Romans, and leadership of the church, feared the druids after they were assimilated into Roman culture. It is stated that it was common knowledge that druids were wielders of magic and able to cast spells.
I have theorized that the church suggested that druids could shapeshift into animals due to the druids affinity to magic and the myths that were already present in Celtish mythology. I cannot find a time period where this would have occurred, and cannot find any scholarly references that would back up or dispute this theory. Hence the question.