Yes, it use to be that the senior apostle would be the next President of the church. This happened with Brigham Young, John Taylor, Wilford Woodruff, and Lorenzo Snow. They were all the Senior Apostle, and they all became president of the church. But something happened on March 31, 1900 to change this.
It turns out that Brigham Young, in 1855, ordained his 11 year old son,John Willard Young , as an Apostle. You can speculate all you want on why he would do this, but to me it looks like he was laying the path for a Young family dynasty of church leadership.
John Young served as a counselor to his father for several years, and as a “Counselor” to the Quorum of the 12, but he was never in the Q12. He live a lot of time in New York City, and made some bad investments. He was accused by Joseph F. Smith of using church funds to support a rather lavish lifestyle, and the Q12 was looking to release him as a counselor when he resigned.
Nine years later, when it became apparent that John was the 2nd most senior apostle behind Lorenzo Snow, the First Presidency, fearing a John Young presidency would be a disaster for the church, changed the rules of succession from senior apostle to senior member of the Q12. And it has been like that ever since.
(For the most part this is just semantics, as I know of no other non Q12 Apostle that was even close to being the senior apostle, and the senior apostle has been the senior member of the Q12 ever since then, but rules are rules!)
For a more thorough treatment of this and other changes to the succession rules, see this Interview with Greg Prince on the Gospel Tangents podcast.
Wheat & Tares 
Brigham did also ordain two more of his sons: Joseph Angell Young and Brigham Young Jr. BY Jr. did actually serve in the Q12. When he died, he was the president of the 12. So, he was close to being the prophet. However, he was ordained apostle before Joseph F. Smith, so according to the date of ordination BY Jr. would have been his senior. But according to the time in the quorum Joseph F. Smith was the senior and he became the prophet.
Todd Compton has an article about John Willard Young and the succession:
Click to access Dialogue_V35N04_125.pdf
The OP’s first paragraph is an oversimplification. It seems there were changes in “seniority” or its definition that affected succession. “The first change occurred in 1861, reversing the seniority of John Taylor and Wilford Woodruff. The second change occurred in 1875, making Taylor and Woodruff senior to two original members of the Quorum of the Twelve, Orson Hyde and Orson Pratt. The final change occurred in 1900, making Joseph F. Smith senior to Brigham Young, Jr.” https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=1370&context=etd
The is no scriptural basis for the succession-based-on-seniority practice. This seems to have been well-understood at least by the Q12 in mid-20th century when they considered changing it, instructed Deseret News to prepare two different front pages declaring two different new Church presidents and then did not convey a decision which to use until mid-morning. Source: conversation with a journalist employed at Deseret News at the time.
While I do not expect any variation from 20th century practice in the current case, it would seem that one cannot rely on practice as the determining factor and still think the specific call is a matter of revelation from God without concluding that one mode of revelation is God’s causing apostles to die in His chosen order.* I prefer thinking of the practice as a matter of coincidence, however intentionally naive that may be. Rules are not necessarily rules. The succession “rule” serves a useful purpose, but I hate to think of the workings of the Spirit or revelation from God being confined by organizational rules. There’s too much evidence elsewhere that they are not.
*It has been suggested by a friend, in the alternative, that God doesn’t need to be involved in bumping off apostles, because any of them could be a Church president acceptable to Him, the revelation happening only in connection with their calls to the Q12 (and possibly God’s foreknowledge).
While serving a mission in 1992 I was interviewed by Robert Wells then a member of the Q70. At one point he was the longest serving GA to never be in the Q12.
He told me that when new men are ordained apostles (and presumably to the Q12) that part of the prayer wording is that when they become the most senior apostle because of death of other apostles they automatically become president if the church. At the time it sounded automatic and instantaneous. That’s not to say it’s always been that way or that it hasn’t changed since.
Toad, the articles of incorporation of the Corperation of the FP say the same thing that’s Elder Wells said. Upon the death the the Corperate Sole (President), the next senior member of the Q12 automatically become the new President.
BTW, I too spent time with Elder Wells in Chile on my mission. He once brought his teenage daughter with him and all us Elders thought she was quite good looking. Little did we know she was a future Miss America!
As I read the amended articles of the Corporation of the President they do not say the next senior member of the Q23 automatically becomes the new president. Instead they allow the Q12 to appoint any one of their number pending installation of a new President of the Church. See article 4 of the amended articles:
“Fourth: The title of the person making these articles of incorporation is “President of the Church of Jesus Christ of Latter-day Saints.” He and his successor in office shall be deemed and are hereby created a body politic and corporation sole with perpetual succession, having all the powers and rights and authority in these articles specified or provided for by law. But in the event of the death or resignation from office of the President of the Church of Jesus Christ of Latter-day Saints, or in the event of a vacancy in that office from any cause, the President or Acting President of the Quorum of the Twelve Apostles of said Church, or one of the members of said Quorum thereunto designated by that Quorum, shall, pending the installation of a successor President of the Church of Jesus Christ of Latter-day Saints, be the corporation sole under these articles, and the laws pursuant to which they are made, and shall be and is authorized in his official capacity to execute in the name of the corporation all documents or other writings necessary to the carrying on of its purpose and to do all things in the name of the corporation which the original signer of the articles of incorporation might do: it being the purpose of these articles that there shall be no failure in succession in the office of such corporation sole.”
But perhaps that internet source is not up to date. I have not requested a copy of the current articles from the Utah Secretary of State.
JR, I see what you are saying. It appears that the 12 can pick anybody they want to be the Corperate Sole. I guess the takeaway is that there was is somebody (today) that is the Corperate Sole for the Corperation, even before a new president is announced. It would be Nelson unless the 14 picked somebody else. This looks like an interim position until the 14 select a new president.
Yes. Sorry about the “Q23” typo. I also noted in an old Ensign article GBH’s description of selecting and ordaining a new president of the church. It appears that this was done by the Q12 (14) in their meeting and likely without the 70s present. I suspect the ordination of new apostles is similar. If that is correct, I wonder what Robert Wells source was for his reported comment on the content of that ordination to the apostleship and, if he were correctly reported and right about that ordination, why the ordination is inconsistent with the legally governing document.
Orson Pratt was also the most senior apostle and should have been church president. They also changed the rule for uninterrupted service as he was briefly out of the Church in Nauvoo.
If the 14 are reading this to make sure they do it properly, can I recommend they read the SS lesson for today where it describes revelation. Speaking face to face with God. And describes that when Moses, who was a man in his prime, afterwards he fell to the earth and didn’t recover for many hours. How young do you have to be to survive this experience?
From the descriptions of what the current men think is revelation, it is really inspiration, which is the promptings of the HG filtered through ones prejudices.
We need to know by revelation, what God wants; not filtered through prejudices. The Apostles had to overcome their prejudice in 1978, we need the same thing om how we treat women and gays now. We need a Prophet willing, and capable of speaking to God face to face. That could have some credibility.
Pres. Lorenzo Snow saw Christ and he lived to tell about it!
“The Apostles had to overcome their prejudice in 1978, we need the same thing om how we treat women and gays now. We need a Prophet willing, and capable of speaking to God face to face. ”
I agree 100%. However, we’d have to be willing to accept what He says e.g. like Snuff who got a face to face revelation that women won’t hold the priesthood but do have some veto power over it.
I stayed with Elder Wells and his family a short time in 1978, in his beautiful home in Buenos Aires, while on my mission in Argentina where he served as a Seventy. I got to know him and his family. He was a godly man with a strong Spirit about him. His children were still quite young; so the beautiful daughter you mentioned who later became Miss America (I didn’t know that) was a little girl when I met her.