Without Gollum's intervention Sauron's reaction time would have been fast enough to have prevented the ring from being destroyed.

In a 1963 letter written in response to a reader's inquiry about Frodo's failure to surrender the Ring in the Cracks of Doom, Tolkien discussed what would have happened had Gollum not been there to instantly steal and destroy the ring after Frodo claimed it.

He explains that Sauron's reaction when Frodo claimed the Ring was to sent the Nazgul to Mount Doom, where they would feign allegiance to Frodo as the new owner of the Ring, with the purpose of getting him outside of the Crack so they could destroy the entrance and/or stalling for time until Sauron could get there and defeat Frodo 1v1.

When Sauron was aware of the seizure of the Ring his one hope was in its power: that the claimant would be unable to relinquish it until Sauron had time to deal with him. Frodo too would then probably, if not attacked, have had to take the same way: cast himself with the Ring into the abyss. If not he would of course have completely failed. It is an interesting problem: how Sauron would have acted or the claimant have resisted. Sauron sent at once the Ringwraiths. They were naturally fully instructed, and in no way deceived as to the real lordship of the Ring. The wearer would not be invisible to them, but the reverse; and the more vulnerable to their weapons. But the situation was now different to that under Weathertop, where Frodo acted merely in fear and wished only to use (in vain) the Ring's subsidiary power of conferring invisibility. He had grown since then. Would they have been immune from its power if he claimed it as an instrument of command and domination?
Not wholly. I do not think they could have attacked him with violence, nor laid hold upon him or taken him captive; they would have obeyed or feigned to obey any minor commands of his that did not interfere with their errand – laid upon them by Sauron, who still through their nine rings (which he held) had primary control of their wills. That errand was to remove Frodo from the Crack. Once he lost the power or opportunity to destroy the Ring, the end could not be in doubt – saving help from outside, which was hardly even remotely possible. Frodo had become a considerable person, but of a special kind: in spiritual enlargement rather than in increase of physical or mental power; his will was much stronger than it had been, but so far it had been exercised in resisting not using the Ring and with the object of destroying it. He needed time, much time, before he could control the Ring or (which in such a case is the same) before it could control him; before his will and arrogance could grow to a stature in which he could dominate other major hostile wills. Even so for a long time his acts and commands would still have to seem 'good' to him, to be for the benefit of others beside himself.
The situation as between Frodo with the Ring and the Eight (The Witch-king had been reduced to impotence.) might be compared to that of a small brave man armed with a devastating weapon, faced by eight savage warriors of great strength and agility armed with poisoned blades. The man's weakness was that he did not know how to use his weapon yet; and he was by temperament and training averse to violence. Their weakness that the man's weapon was a thing that filled them with fear as an object of terror in their religious cult, by which they had been conditioned to treat one who wielded it with servility. I think they would have shown 'servility'. They would have greeted Frodo as 'Lord'. With fair speeches they would have induced him to leave the Sammath Naur – for instance 'to look upon his new kingdom, and behold afar with his new sight the abode of power that he must now claim and turn to his own purposes'. Once outside the chamber while he was gazing some of them would have destroyed the entrance. Frodo would by then probably have been already too enmeshed in great plans of reformed rule – like but far greater and wider than the vision that tempted Sam (III 177) – to heed this. But if he still preserved some sanity and partly understood the significance of it, so that he refused now to go with them to Barad-dûr, they would simply have waited. Until Sauron himself came. In any case a confrontation of Frodo and Sauron would soon have taken place, if the Ring was intact. Its result was inevitable. Frodo would have been utterly overthrown: crushed to dust, or preserved in torment as a gibbering slave.
_{September 1963 Letter to Mrs Eileen Elgar}

In the book itself we see a brief hint of this, as we see the Nazgul leaving the battlefield as soon as the ring gets claimed.

But the Nazgûl turned and fled, and vanished into Mordor’s shadows, hearing a sudden terrible call out of the Dark Tower; and even at that moment all the hosts of Mordor trembled, doubt clutched their hearts, their laughter failed, their hands shook and their limbs were loosed. The Power that drove them on and filled them with hate and fury was wavering, its will was removed from them; and now looking in the eyes of their enemies they saw a deadly light and were afraid.
_{The Lord of the Rings - Bok VI Chapter 4 - "The Field of Cormallen"}