It wanted to stay hidden until the proper time
Most of this is synthesis on my part, as there are no detailed writings concerning the Ring's intentions.
If the Ring is a mostly dumb object, the answer could be that it didn't plan ahead: "I am on Isildur's finger, Isildur is not my master, therefore I must slip". It could also have been a random event, as Bilbo mentions that it "might suddenly slip off a finger where it had been tight".
If the Ring is less dumb, it could be that it wants to hide because Sauron is too weak. It only reappears when Sauron becomes strong enough to call for it.
However, if we assume the Ring is smart and can make decisions on its own, then we can come up with a couple of reasons.
A Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it.
The Lord of the Rings: The Fellowship the Ring, The Shadow of the Past, p. 73
Sauron is gone
Sauron's physical form was destroyed at the end of the War of the Last Alliance by Elendil, father of Isildur, and Gil-galad.
Then Sauron was for that time vanquished, and he forsook his body, and his spirit fled far away and hid in waste places.
The Silmarillion, Of the Rings of Power, p. 353
He would not reappear for a thousand years.
It's not clear what would happen if you put the Ring next to his disembodied spirit. Because the Ring is a physical artifact, a body could be required to use its powers. Sauron did bring back the Ring from Númenor in spirit form after the island's destruction, but whether it helped him get a body back is unclear to me.1
Assuming he can't use it, it is useless for the Ring to immediately try to get back to Sauron. If Sauron could use the Ring, then it may be that it doesn't know where he is or how to reach him. In both cases, secrecy would be more important.
Isildur doesn't trust it anymore
Isildur is coming to his senses and wants to give the Ring to the "Keepers of the Three". At this time, this was Galadriel, Círdan and Elrond:
"I cannot use it. I dread the pain of touching it. And I have not yet found the strength to bend it to my will. It needs one greater than I now know myself to be. My pride has fallen. It should go to the Keepers of the Three." [...]
"My King," said Elendur, "Ciryon is dead and Aratan is dying. Your last counsellor must advise nay command you, as you commanded Ohtar. Go! Take your burden, and at all costs bring it to the Keepers: even at the cost of abandoning your men and me!"
Unfinished Tales, The Disaster of the Gladden Fields, p. 354-355
The Ring would slip from his finger minutes later. Getting picked up by an Orc or even being lost in the river would be a better fate than ending up with some of Sauron's most powerful enemies.
Others want to destroy it
There are many people who are aware of Isildur having the Ring. Among them are Elrond and Círdan, two of the most powerful Elves at the time, who urged Isildur to destroy the Ring. The longer the Ring stays with him, the more likely it is that someone will take it and attempt to destroy it. At the time, Mordor is basically destroyed after having been under siege for seven years. The journey to Mount Doom would be a walk in the park.
"But few marked what Isildur did. He alone stood by his father in that last mortal contest; and by Gil-galad only Cirdan stood, and I. But Isildur would not listen to our counsel." — Elrond
The Lord of the Rings: The Fellowship of the Ring, The Council of Elrond, p. 319
It doesn't want to make someone else too powerful
If Isildur is able to keep the Ring, he could end up being a very powerful man. He is not on the level of a Maia like Gandalf, but he is still a Númenórean at a time when they are at their most powerful: a little more than 100 years before, an armada of ships from Númenór had invaded Valinor.
Isildur being in full possession of the Ring could mean that he would become too powerful even for Sauron, in the event of his return. It also means he would likely become a subject of intense scrutiny from anybody else in a position of power. If knowledge of the Ring and its powers were to become public, it might have even more trouble returning to Sauron.
Why did it wait so long?
Sauron is first seen taking a physical form around the year 1050 of the Third Age, a full millennium after Isildur's death. He takes over Dol Guldur in the forest of Mirkood. The Ring might have thought of leaving its hiding place, but the Istari were sent by the Valar at the same time. The Ring might have felt that Sauron was not in a position where it would be safe.
And [Saruman] deemed that the Ring, which was Sauron's, would seek for its master as he became manifest once more; but if he were driven out, then it would lie hid.
The Silmarillion, Of the Rings of Power, p. 362
Indeed, Sauron would wage war by proxy, using the Nazgûl to destroy the northern kingdom of Arnor and capture Minis Ithil, which would become Minas Morgul. In 2063, Gandalf would drive Sauron, who was still only known as the Necromancer, out of Mirkwood. Had the Ring worked its way towards Sauron, it might have been found or perhaps Sauron would not have been strong enough to protect it against the Istari.
Out of hiding
It was only 400 years later, in 2460, that Sauron would come back in strength to Dol Guldur. At the same time though, there was a lot of activity in the Gladden river:
[Saruman] set a watch upon the Gladden Fields; but soon he discovered that the servants of Dol Guldur were searching all the ways of the River in that region.
The Silmarillion, Of the Rings of Power, p. 362
The Ring might have decided that its master was ready or perhaps it became concerned that it might be found by the wrong people. It may have been forced to act quickly. Three years later, two Stoor Hobbits by the name of Sméagol and Déagol would find it.
The rest, as they say, is history.
1 Sauron only needs 100 years after Númenór's destruction to get his new body and fight in the War of the Last Alliance. After losing the Ring, he requires 1000 years to first reappear in Mirkwood as the Necromancer.
I'm not sure why there's a difference, but maybe there isn't actually one: "Sauron was, of course, 'confounded' by the disaster[...] He was attacked by Gil-galad and Elendil before his new domination was fully established." (Letters #211) So it may be that he would have needed 1000 years after Númenór and that, Ring or not, Sauron generally needs a lot of time to get a new body.
All citations from the Harper Collins editions