If the other driver is at fault in a collision and does not have insurance can the claim be paid entirely under your uninsured driver coverage?

Yes - that's the point of uninsured motorist coverage. You'll be bound by the terms of your policy so review it. But generally yes, if the at-fault driver is uninsured, you can make a claim against your uninsured motorist policy. You'll have to prove the other driver had no insurance. Generally an affidavit of no insurance, or a letter from the most recent insurer stating the policy had lapsed or was in some way not in force is sufficient.