A lot depends on how far back you want to assume this change of heart goes.
Based on the ROTK appendices, Saruman began searching the ring at least 150 years before the the events in the LOTR, and much of his counsel in the White Council was shaded with the hopes of finding the ring himself. So perhaps he would have supported Gandalf’s call for an attack on Dol Goldur. This would have been about 90 years before the actual attack took place (at the time of the events of the Hobbit).
Perhaps this is early enough that Sauron is still weak and had not rebuilt enough strength in Mordor and so he might have been dispersed to the East for another few hundred years of a “watchful peace.” Is this good? Perhaps.
But Bilbo is about to find the ring and bring it back to the surface. Or maybe not? Maybe Gandalf is not so concerned about Smaug since the Necromancer is no longer haunting Mirkwood, or perhaps he’s chasing rumors of the shadow further to the east. Or maybe the events of the Hobbit still take place, but afterwards Gandalf doesn’t keep as close an eye on Bilbo and Frodo since he is no longer worried about rings and shadows and Dark Lords, and so the ring causes some new kind of mischief. Or perhaps it gets forgotten about in some hobbit hole for a few hundred years until Sauron arises again.
And maybe this future time will find Sauron arising when the strength of the elves has further diminished, and the kingdoms of men are more weakened and divided. And so the land of Middle Earth will be more vulnerable to his conquest.
But I’m going to ignore all that, and assume the history of Middle Earth is essentially unchanged until the start of the LOTR. In that case I think BeepKillBeep is right. The journey for Frodo from the Shire to Rivendell should be much easier since, without being imprisoned in Orthanc, Gandalf can reach the Shire early and the hobbits can leave in July with Gandalf as a companion instead of in September on their own. And maybe if Saruman in Orthanc is opposed to them the Black Riders might not be able to cross the Fords of Isen, or at least they might be slowed down.
I’m not sure what happens to the Council of Elrond if Frodo arrives in Rivendell in August instead of October. Is everyone else there? Is Saruman there? Does Saruman become a member of the Company of the Ring?
In any case the journey of the Fellowship should be much easier. Without Saruman’s mischief Dunland is much safer and the company can take Boromir’s advice and travel through the Gap of Rohan, perhaps resting in the safety of Orthanc instead of Lorien, and then continue through Rohan to Gondor, and then plan their infiltration of Mordor from there.
So what has changed?
No Moria. So the Balrog is not killed and there is no Gandalf the White. Is this good? Maybe not, the Balrog might cause trouble elsewhere, maybe an attack on Lorien. No Gandalf the White could certainly be a problem later on.
No visit to Lorien and no looking into Galdriel’s mirror. Also no gifts from Galadriel. The mirror is maybe an indifferent change, but the elven cloaks, lembas, rope, and particularly the light given to Frodo all proved to be of great worth once in Mordor.
No splitting of the Fellowship. No death of Boromir. Likely no reason for Aragorn to travel the paths of the dead. Does Aragorn use the palantir of Orthanc, revealing himself to Sauron and upsetting his plans? What happens when Boromir, Gandalf, Aragorn, the broken sword and the bane of Isildur all come knocking at the gates of Minas Tirith? How does Denethor react? Will the ring bearer be allowed to leave Gondor? If so, does Gandalf travel with them? Is that a good idea? Part of the success that Frodo and Sam had was being able to sneak about almost unseen. Would a powerful wizard like Gandalf be able to sneak into Mordor without raising any alarms? Who knows?
No Ents? The Battle of Pelennor Fields? No army of the dead? No Aragorn with ships from the South? No Gandalf at Minas Tirith during the battle? I’m sure I’m missing all other potential changes.
So no betrayal by Saruman has many obvious good effects. But there are so many unseen follow on effects and unintended consequences that it’s impossible to tell what the net effect would be. The whole story of the LOTR is built on happenstance, good fortune, and happenstance that even tiny changes could change the whole timeline.
Edit And No Tom Bombadbil!!! What would happen to the story then?