Obviously not. Although it appears that the Nine probably rested with Sauron, and were probably not carried around by the Nazgul themselves, the Rings were not merely responsible for making the Ringwraiths 'appear'; the reason that they stayed devoted to Sauron was their unbreakable connection with him through the Nine.
While the Rings themselves were with Sauron (we don't actually know this for sure, by the way), each Ring was stilled 'owned' by a Nazgul. They were not designed for more than one wearer.
Were Sauron somehow to do this, I would guess that each Nazgul or neo-Nazgul would be quite substantially weaker than they were with the normal one Ring per wraith. The sustained power would have to be dissipated amongst them all, of course. But the Nine Rings did not work this way; one owner, and total devotion with that owner -- and of course with the Lord of the Rings, Sauron, with whom by the Rings a channel to each Nazgul was maintained.
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