The One Ring and Sauron

[Gwaihir the Windlord]

That would mean his "soul" or "fea" ceased to be, and he would then be outside Arda. Saruman would probably go to Namo...

If his fea ceased to be, then he would probably not actually be anywhere... At any rate this could not have happened. Fear cannot be destroyed.

He would remain in the world, as the Ainur can never be truly killed within Arda (they tend to hang around, until they are strong enough to take up a physical form again and rise), but -- as the Ring, into which he had poured much of his energy -- was now destroyed, he was too weak to ever again forge himself a new hroa. A mere spirit of mist, blown about by the winds of Middle-Earth is what he became -- unless he indeed left Arda and entered the outer halls of Ea (space, basically), in which case he would be blown about by the solar winds up there forever. Perhaps he found his way back to Melkor, and there resides with him as his dark lord slowly gathers strength, rebuilding his power for his last strike on the world...

Sauron could not have left Ea, though. The Ainur that entered the world at the beginning were bound to stay there until its end.

As for Saruman, I do not think that Mandos took him -- at least not until he had been left in Middle-Earth for a bit.

To the dismay of those that stood by, about the body of Saruman a grey mist gathered, and rising slowly to a great height like smoke from a fire, as a pale shrouded figure it loomed over the Hill. For a moment it wavered, looking to the West; but out of the West came a cold wind, and it bent away, and with a sigh dissolved into nothing.

It looks as if the Valar rejected him. Whether or not he was able to later take incarnate form, I should say he was doomed to stay in Middle-Earth.