Two things to remember here - Sauron didn't know about the Fellowship, and furthermore could never have imagined that the Ring would be destroyed rather than used against him by the Free Peoples.
If Sauron took the Ring, the war would be over fairly soon afterward - that much is clear. So, instead of sending nine riders north who would be of far greater value leading his armies, I would have instead told Saruman to march his armies north, to ravage the Shire and find the Ring. No army in Eriador could counter ten thousand Uruk-hai and wildmen (though Rivendell could probably hold out against them). The Ring would then have most likely been found.
If the Company of the Ring had been formed and they had gotten out of Eriador (and therefore Saruman's armies were in Eriador and none able to ambush them) they wouldn't have been broken at Amon Hen. This is pure speculation, but I imagine that Aragorn would have pressed them on to the Morgul Vale, rather than listening to Boromir's insistence that they go to Minas Tirith. By this point, Sauron's armies would be besieging the city, so it probably wouldn't be the best idea anyway.
This is where it gets interesting.
Gandalf would have returned, yes, but where? My guess is that he would have returned to the Company in Ithilien (as they were the only real hope of defeating Sauron), and they then with him would have gone through Shelob's lair (Aragorn, Gandalf, Legolas, Gimli and Boromir probably making short work of the beast if they encountered her).
However, they would have great difficulty in getting across Gorgoroth to Mount Doom, as Sauron did not put forth all his forces to attack Minas Tirith (which probably would have fallen by this point without Gandalf rallying the defences, and indeed without Merry to provide the crucial blow to the Witch-king). The Fellowship would have had little choice however, so they would have pressed on.
I find it fairly unlikely that they could have got to Mount Doom from Cirith Ungol without being spotted, and probably would have been overwhelmed by orcs. However, beyond that point, there is nothing more Sauron could have done. Again, it never entered Sauron's wildest dreams that the Ring wouldn't be used against him in some way, and so guards placed on a place none of his enemies would ever have any want or need to go would have seemed foolish to him.