Well, for a start, what are the Nazgul doing in the shire with no one ring to pursue? Scratch that, with no rings of power, the Nazgul do not exist.
Assuming the one ring was destroyed at the end of the second age, as it should have been, there are a number of consequences. I’ll assume that Sauron survived because he placed less of his essence into the one ring, so that while diminished, he retains some of his powers. No disaster of the Gladden Fields, so the dunedain of Arnor are strengthened. No Witch-King, no Angmar, so we can be pretty sure Arnor would have survived and prospered. No thain, as there is still a king in Arnor. Minas Ithil would probably still be in Gondor’s hands, with no ringwraiths to lead the siege against it. With Ithilien still occupied, Gondor has some extra manpower to draw on. Sauron might have a fight on his hands when re-occupying Mordor. A second fortress also helps protect Gondor from attacks from the Easterlings and Haradrim. Any army advancing far into Gondor risks being attacked in the rear and being cut off. It also makes it harder for Sauron to mass his forces, no Haradrim wandering up to the black gate. Finally, no Saruman to threaten Anorien/Rohan.
On the other hand, Sauron would have started to take shape earlier in the third age, being less dependent on his ring, so his plans might be further advanced. Without Galadriel’s ring to protect it, Lothlorien might well be deserted, many elves fled when the Balrog awoke in Moria. The population of Rivendell and Lindon is diminished, more elves have wearied of and left middle-earth. No Gandalf, but in this scenario he isn’t as crucial, it’s pretty much a straight military campaign.
Sauron still has numbers on his side, with the population of the east and south at his disposal, and orses, lots of orcses. He is robbed of his great terror weapon, the Nazgul. Reading the account of the siege of Minas Tirith, without them there will be no quick victory over a fortress. This makes dealing with Gondor a real problem, a besieging army is vulnerable to counter-attack. He needs to take Minas Ithil first, as his forces can be safely (well, Shelob can only eat so many) supplied down the pass of Cirith Ungol.
All this is subject to some serious butterfly effect. For example, a stronger Gondor may never have ceded Anorien to the Rohirrim. A lot depends on how co-ordinated the forces of the west are. If they are isolated, Sauron can grind them down and pick them off one at time. If they are united, they may be able to hold off Sauron’s forces indefinitely.