Would they become wraiths and dominated by Sauron, or would they just be able to be mind-read?
Sauron would see everything that had been done by the power of the Elven Rings and gain control of the wearers.
And while he wore the One Ring he could perceive all the things that were done by means of the lesser rings, and he could see and govern the very thoughts of those that wore them. But the Elves were not so lightly to be caught. As soon as Sauron set the One Ring upon his finger they were aware of him; and they knew him, and perceived he would be master of them, and of all that they wrought. Then in anger and fear they took off their rings.
Of the Rings of Power and the Third Age - The Silmarillion
So it would appear that Elves -- certainly Celebrimbor of Eregion, and seemingly one or more of the other Elven-smiths as well -- were wearing the Elven Rings for a brief instant when Sauron first put on the One Ring. Unfortunately for Sauron, the process also worked in reverse for them: Celebrimbor at least understood exactly what had happened and hid the Three Rings before Sauron and his army came to Eregion to take all the Rings by force.
Per Gandalf, the process of becoming a wraith is tied to use of a Ring:
And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the dark power that rules the Rings.
The Shadow of the Past - The Fellowship of the Ring
If the wearer has control of the Ring (rather than vice versa), it appears becoming invisible is an at-will power. Galadriel is wearing her Ring in "The Mirror of Galadriel" but is, of course, quite visible to Frodo and Sam. Likewise, Tom Bombadil dons the One Ring and it does not automatically turn him invisible as it does the hobbits. Likewise, Sauron's possession of the One Ring would not, in itself, accelerate the process of turning the owners of the Elven Rings into wraiths -- though being under his mental control wouldn't be a huge improvement...