Many countries have continuity of government plans for what to do if the leadership suddenly dies in some sort of a catastrophe. The United States famously has a “designated survivor” that is in a secure location during the State of the Union so that somebody in the line of succession can be preserved if the Capitol building is destroyed; one of the of the most interesting manifestations of this planning is the fact that each of the four UK nuclear missile submarines has a handwritten letter from the Prime Minister in a safe that has instructions for what to do with their nuclear missiles in case the UK government is destroyed.
The hierarchy of the Church is structured around the doctrine that the top fifteen men in the Church have the same office as Christ’s original apostles, and that they are ordained to that position by other apostles, eventually stretching back to Jesus Christ through the laying on of hands. But what would happen if all fifteen pass away at the same time?
The lack of a clear, legal successor might lead to a post-Joseph Smith martyrdom situation, with various splinter groups and claimants. Institutionally it seems the most likely scenario would be that the Presidents of the 70s would become the new governing body of the Church (assuming they were not affected by the catastrophe). D&C 107:26 could be interpreted as meaning that, just as the Quorum of the 12 becomes the governing body when the First Presidency dissolves, so to do the Quorums of the 70 become the governing body of the Church if the Quorum of the 12 dissolves. While the Presiding Bishopric might also be a natural option, I assume that the fact that theirs is an Aaronic priesthood office removes them from the line of Melchizedek priesthood succession. However, I’m not much of a Latter-day Saint “canon lawyer.” and others might have more informed takes on what, canonically, could be done in that situation, and what the implications are.
For example, would the apostolic keys be passed to the new governing body (whatever that would be)? Or would the re-establishment of the apostleship require another heavenly visitation and ordination? The chance of the worst case scenario happening, while thankfully small, is still not negligible; since the fifteen prophets presumably often meet together, the bench is not that deep. (I heard that there is a “designated survivor” arrangement at General Conference, with one of the fifteen not in attendance in each session, but that might just be a rumor and I haven’t seen a solid source on it.)
If the laying on of hands from a previous apostle is taken as non-negotiable for apostolic continuity it’s hard to know what measures could be taken outside of basic security precautions to assure that the line is preserved. One outside-the-box possibility stems from the fact that apostleship is not synonymous with membership in the Quorum of the 12. For example, Brigham Young non-publicly ordained his son John to the apostleship while not placing him in the Quorum of the 12. In theory a non-Quorum of the 12 general authority could be ordained an apostle (publicly or otherwise) while still serving as a 70 in Timbuktu, and this would help preserve the apostolic office in a worst case scenario. (In the case of John Young, he eventually moved to New York City and became a tragic, broken figure; however, if a catastrophe had happened to the Utah-based apostles, then in theory John Young would have been the sole ordained apostle left). I’m just speculating here; it’s not my business to tell the Church what to do, but whatever the case I hope that the powers that be have given serious thought to a theological and administrative continuity of government scenario even if the answer is “nothing while we wait for more celestial visitors.”