Given the triune God, one nature and three persons, do they experience their will in their singular nature or in their separate persons? That is, do they share one will, or does each have a separate will which is in perfect harmony with the others?
While I'm not 100% certain I understand your question. If you're asking what I think; Jesus implied that the will of the Father, Son, and Spirit are separate, as He spoke a lot about the Father's will.
So I think Jesus clearly implied that He had the ability to disagree, but He chose to do the will of the Father. I think the clearest statement of this is in John 8:27-29, NIV
While the answer from WhatAboutJohn3_17 is fantastic, I thought I would at least add some more resources, to help give more depth.
I think the difficulty is trying to think of the Father, the Son or the Holy Spirit as existing separately from God.
This would be related to the question about the person of God, in relation to the Trinity, as explained by St. Thomas Aquinas:
God is a self-subsistant person that embodies all that is perfect, so though we label them separately, they are one.
To see if they have equality and what their likeness is like you can look at
but, basically, they are co-equal in all respects, so the Father is not over the Son.
Then you may want to look at the essence of the three persons:
Though it isn't spelled out in Scripture that God and the Son have the same essence Jesus did state "The Father and I and the Father are one (John 10:30)", for example.
So, the God is a person and has the same essence, and the person and property of God is the same (http://www.newadvent.org/summa/1040.htm), then, though this is a bit long-winded, but since God is a single person then we can see that though the Trinity may be separate, we can't differentiate between the will of each and the Will of God, as our sense of three does not force God to be that way (as explained more below). To see more about the Will of God you can read http://www.newadvent.org/summa/1019.htm.
You may want to read about the plurality of God (http://www.newadvent.org/summa/1030.htm) especially Article 3, where this comment is made:
So, the idea is that when we talk about three persons, that does not signify anything about God in reality. We have one God, and though, for our sake we express God as three persons, that is still for our simplicity, it does not mean that we have three gods that are one God, so, God will have one will from God, as, just because we think of God as three person doesn't mean that that changes what God is.