First, I'm sure that Presidents Eyring and Uchtdorf get along great, but in this condition we have two "masters". There cannot be two masters as the scripture quoting Jesus informs us. Though this is a temporary, yet uncertain as to the duration, situation, progress in the Church is slowed, for because Pres. Monson is still alive and not replaced, neither of these two Counselors can receive commandments and revelations for the Church, according to Doctrine and Covenants 28 and 43.
Secondly, with the death of three Apostles in a row in 2015, President Monson secretly made a change to the Quorum of 70 by joining the First and Second Quorums into one General Authority Seventy. In Doctrine and Covenants 107, the Quorum of 70 can act when there is no functioning Quorum of 12 and likewise when there is no functioning First Presidency, the Quorum of the 12 have authority to act. But they must be in 100% agreement in all decisions or nothing can be done; and there must be a majority to conduct business. Thus if there were a scenario where there was no functioning First Presidency and less than 7 members of the Quorum of the 12, the Quorum of 70 would then be the Head of the Church. But then if in that hypothetical situation there were less than 36 members of the Quorum of the 70, then there would be no functioning leadership of the Church, though the Presidency of the 70 have authority to appoint other 70; thus the 70 Presidency would also have to be non-functioning. Because there is a way for this to happen, a contingency must be made, for the longer we go with the President incapacitated the more likely such a scenario could play out.
Elder Paul H. Dunn of the Seventy was the first General Authority to be placed on Emeritus status. This is the Church's version of retirement, though they could still function in another calling. In the Quorum of the 12 there have been situations were there was an Acting President, while the ones who would have been President were in the First Presidency. And President Hunter is the first and only Apostle to become the Acting President of the 12 when President Marion G. Romney was incapacitated due to illness.
So I present three solutions that would require setting new precedent in the Church:
#1 - The First Counselor in the First Presidency is appointed by the 12 as the Acting President of the Church. He therefore can appoint new or more counselors as needed.
#2 - The 12 place President Monson on Emeritus Status, which would then place President Nelson as the new President of the Church.
#3 - The 12 place President Nelson as Acting President of the Church, until his or Monson's death or miraculous recovery.
As I referred, the 12 must be in 100% agreement with whichever of these options they select. With the deaths of all the 12 Apostles of Jesus Christ before replacements could be made, Christ's Church lost authority and fell into Apostasy. We are promised by the Lord in revelation to Joseph Smith that the keys of authority will never again be taken from the Earth, but we don't want to jeopardize the fulfillment of the revelation.
Personally, I prefer #'s 2 or 3, more so with number two, though what do you do with a miraculous recovery? If the new Egyptological research is true that Moses was a Pharaoh of Egypt and came back to take back the stolen throne, #2 could see that same type of hypothetical scenario. Unfortunately it is the nature and disposition of man to not act until it becomes necessary to act, but in this case however, prevention would be the wiser course of action. There was one day in which my website had 6 individuals who were new to my site, all on the same day looked at all of my posts at that time. The stats stood out very clearly from my regular pattern of viewers. If that was 6 members of the Quorum of the 12, then I'm sure you are still occasionally checking in on my site, thus I have made my presentation and have fulfilled my assignment. The matter is now in your hands.