Lord President-elect

Lord President-elect was the title given to the President-elect of the Time Lord presidency. (TV: The Invasion of Time)

The Seventh Doctor once introduced himself to Davros who was acting as Dalek Emperor at the time as "President-elect of the High Council of Time Lords, Keeper of the Legacy of Rassilon, Defender of the Laws of Time, Protector of Gallifrey". (TV: Remembrance of the Daleks) The Eighth Doctor later told Sam Jones "about being President Elect of the High Council of Time Lords, keeper of the Legacy of Rassilon, Defender of the Laws of Time and Protector of Galloway. Or something." (PROSE: Unnatural History)

The existence of the position was exceedingly rare and for an extended period of time even more so. The incumbent Lord President usually named their successor on Presidential Resignation Day who was sworn in immediately. If this did not happen due to assassination or impeachment, an election would be held and then the victor of that election would be named President-elect, and soon after, be sworn in officially as President. (TV: The Deadly Assassin, AUDIO: Appropriation) The Fourth Doctor was the President-elect for a huge length of time after participating in an election that saw the death of the only other candidate, Chancellor Goth, and then departing Gallifrey. (TV: The Deadly Assassin) He was technically President-elect for the entirety of his travels with Leela and later, K9 Mark I. (TV: The Face of Evil, The Invisible Enemy)

The Gold Usher formally introduced the President-elect to the Matrix, asked the Council if anyone would contest the candidate's right to the office and, eventually, they invested the President and provided to him/her the items of Rassilon. (TV: The Invasion of Time)

Powers
A different individual could not become Lord President if a President-elect existed. When the Fourth Doctor found Gallifrey without a President he claimed the presidency even though some time had passed since the election he took part it. (TV: The Invasion of Time) The President assassinated by Goth directly preceded the Fourth Doctor's presidency. (PROSE: A Brief History of Time Lords) If no other Time Lord attempted to run for the position within a set amount of time, it automatically fell to the President-elect as set out in the Constitution. (PROSE: Doctor Who and the Invasion of Time)

The powers of President-elect was largely the same as that of the Lord President and Acting Lord President. The Fourth Doctor claimed the High Council could ratify nothing even if there was no Lord President and the President-elect was elsewhere. He called their ratification of the appointment of Borusa to the Chancellorship illegal.

The Doctor as President-elect was also able to give orders to Cardinal Borusa, [Castellan Kelner and Commander Andred of the Chancellery Guard. He bestowed the positions of Chancellor-elect and Acting Vice President to Kelner and also ordered for the Presidential Office to be redocarated with lead lined walls, (TV: The Invasion of Time) a choice that remained until the era of President Matthias many years later. (AUDIO: Appropriation, Mindbomb)

History
To escape a death sentence in a trial where he was charged with the murder of the Lord President, the Fourth Doctor invoked Article 17 of the Constitution of the Time Lords to run for President. It was later found that it was Goth, under the the influence of, who committed the crime. As the only other candidate the Doctor was effectively President-elect but left Gallifrey that night. (TV: The Deadly Assassin) When he returned and found the planet without a President, the Doctor claimed the position and was briefly the 407th President. (TV: The Invasion of Time)

Other universes
In a parallel universe where the First Doctor decided not to leave Gallifrey, Susan was President-elect for a time. At the time, the Doctor was experiencing a simulation of Hannibal's crossing the Alps in the Possibility Generator. Together they discovered that "Auld Mortality" was using Badger to regularly hypnotise the Doctor, so he could be manipulated into the position of President. However, realising that Susan was about to become President, he attempted to control her instead. However, the Doctor used the Generator to release Hannibal and his horde on to the Panopticon, and destroyed Badger and defeated Quences, who masterminded the plan. (AUDIO: Auld Mortality) Susan later officially became President. (AUDIO: A Storm of Angels)