Probably between 2851 and 2939, almost certainly before 2953
First, a couple of comments on the question.
Unfinished Tales doesn't say for certain that Saruman found Isildur's remains, only that the original Elendilmir and a chain that might have held the ring around Isildur's neck were found in Orthanc.
In a casket on a high shelf two things were laid. One was a small case of gold, attached to a fine chain; it was empty, and bore no letter or token, but beyond all doubt it had once borne the Ring about Isildur’s neck. Next to it lay a treasure without price, long mourned as lost for ever: the Elendilmir itself, the white star of Elvish crystal upon a fillet of mithril that had descended from Silmarien to Elendil, and had been taken by him as the token of royalty in the North Kingdom.
...
When men considered this secret hoard more closely, they were dismayed. For it seemed to them that these things, and certainly the Elendilmir, could not have been found, unless they had been upon Isildur’s body when he sank; but if that had been in deep water of strong flow they would in time have been swept far away. Therefore Isildur must have fallen not into the deep stream but into shallow water, no more than shoulder-high. Why then, though an Age had passed, were there no traces of his bones? Had Saruman found them, and scorned them – burned them with dishonour in one of his furnaces? If that were so, it was a shameful deed; but not his worst.
Unfinished Tales Part Three, Chapter I: The Disaster of the Gladden Fields
Pages 264-5 (Houghton Mifflin Harcourt; 2012 Kindle Edition)
As Mark Olson suggested in a comment, Tolkien changed his mind about several details over the years. Some of the details in Unfinished Tales had changed by the time that The Lord of the Rings was published.
Saruman began searching for the Ring in 2851.
2851 The White Council meets. Gandalf urges an attack on Dol Guldur. Saruman overrules him. Saruman begins to search near the Gladden Fields.
The Lord of the Rings Appendix B, Section 2: The Third Age
Page 1088 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)
Saruman discovered that Sauron was searching near Gladden Fields in 2939, and agrees to the White Council attacking Dol Guldur in 2941:
2939 Saruman discovers that Sauron’s servants are searching the Anduin near Gladden Fields, and that Sauron therefore has learned of Isildur’s end. He is alarmed, but says nothing to the Council.
2941 Thorin Oakenshield and Gandalf visit Bilbo in the Shire. Bilbo meets Sméagol-Gollum and finds the Ring. The White Council meets; Saruman agrees to an attack on Dol Guldur, since he now wishes to prevent Sauron from searching the River. Sauron having made his plans abandons Dol Guldur. The Battle of the Five Armies in Dale. Death of Thorin II. Bard of Esgaroth slays Smaug. Dáin of the Iron Hills becomes King under the Mountain (Dáin II).
The Lord of the Rings Appendix B, Section 2: The Third Age
Page 1089 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)
The Tale of Years goes on to say that Saruman withdraws to Isengard in 2953.
2953 Last meeting of the White Council. They debate the Rings. Saruman feigns that he has discovered that the One Ring has passed down Anduin to the Sea. Saruman withdraws to Isengard, which he takes as his own, and fortifies it. Being jealous and afraid of Gandalf he sets spies to watch all his movements; and notes his interest in the Shire. He soon begins to keep agents in Bree and the Southfarthing.
The Lord of the Rings Appendix B, Section 2: The Third Age
Page 1089 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)
The earliest date Saruman could have found Isildur's belongings would be 2851 (when he started searching for the Ring). We are not told when he found them or even when he stopped searching for the Ring. However, he must have found Isildur's belongings while he was still searching, and then may well have decided that the ring was no longer in the river (either taken or washed into the sea).
It is likely that he stopped searching at about the time that he became aware that Sauron was searching the same area. If not then, it would probably have been when he withdrew to Isengard.