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Gandalf took years to figure out Bilbo Baggins was in possession of the One Ring in The Lord of the Rings, but he has two good excuses to explain the delay. Bilbo, of course, swipes his magic ring from Gollum during The Hobbit, and it doesn't take long before Gandalf learns about his dear friend's precious acquisition.
Alas, 77 years pass before the wizard declares with certainty that the ring is, in fact, the One Ring forged by Sauron. Gandalf harbors suspicions during The Hobbit, and these grow in the 60-year period before The Lord of the Rings, as Bilbo appears to age more slowly compared to others in the Shire. Bilbo's behavior at his birthday party sparks further alarm, and after investigating the matter in depth, Gandalf returns to the Shire, casts Bilbo's ring into the hearth, and instructs Frodo to get moving.
The Doylist explanation for Gandalf's slowness is simply that J.R.R. Tolkien hadn't yet mapped out Middle-earth's full future while writing The Hobbit (nor precisely what Bilbo's ring would become). Indeed, Tolkien actually rewrote The Hobbit's "Riddles in the Dark" chapter post-publication because the depiction of Gollum and the One Ring didn't fit with The Lord of the Rings.
The in-universe explanation is a little harder to come by, but Gandalf's long road to realization does make sense.
Reason #1: The "Other" Rings In Middle-earth
The Lord of the Rings clearly spells out how many Rings of Power exist in Middle-earth: the Three made by the Elves, the Seven gifted to the Dwarves, and the Nine given to Men. Between The Hobbit and The Lord of the Rings, the Three are accounted for, the Seven are either destroyed or reclaimed by Sauron, and the Nine remain under the Dark Lord's purview.
The wizard also would've known about Sauron's hand in making the Rings of Power, so even if he suspected that Bilbo's could be one of the Seven miraculously recovered, he couldn't have allowed the hobbit to keep it. If these were the only rings in Middle-earth, the first sight of Bilbo having a "magic ring" should be a huge cause of concern for Gandalf.
But Middle-earth is actually home to so-called "lesser" rings that had been produced by the elves as "essays in craft." Crucially, the plain design of Bilbo's ring also gave the outward appearance of being a lower-ranking piece. Given that the nineteen major Rings of Power are more or less accounted for at this point, Gandalf is essentially presented with two options:
Either Bilbo has stumbled across the most powerful item in Middle-earth, or he'd found one of the many lesser rings languishing at the bottom of a mountain - one expressly designed for turning the wearer invisible and pulling pranks. Quite logically, Gandalf draws the latter conclusion, but it speaks to the wizard's intellect that he remains alert to the ring's presence and effects nonetheless.
Reason #2: Saruman Lies About The One Ring
Gandalf might have been more willing to entertain the idea of Bilbo Baggins finding the One Ring were it not for Saruman's meddling. Shortly after The Hobbit, when Gandalf has a slight sense of unease regarding Bilbo, the White Council meets to discuss Sauron's return and the One Ring's whereabouts. Saruman, who was widely accepted as the most learned wizard in matters of ringcraft, assures his allies that the One Ring has fallen into a river and been carried to sea, where even Sauron's minions cannot retrieve it.
Gandalf has absolutely no reason not to trust Saruman. The White Wizard's astuteness, authority, and knowledge of the enemy were thought to be unparalleled, and Gandalf couldn't have imagined the plan of betrayal forming in his fellow maia's mind. Saruman's plan is, of course, to mislead the council, find the One Ring before Sauron, and rise to power. Were it not for these deceptions, Gandalf would have taken a closer look at Bilbo's ring much sooner.
Ultimately, it is Mordor's desperate pursuit of the One Ring, combined with the information he and Aragorn pull from Gollum, that finally pushes Gandalf into seriously considering that Bilbo Baggins has Sauron's prized possession stashed away at Bag End. Only then, decades later, does The Lord of the Rings begin.
