In The Elder Scrolls, was there ever a child that became a King or a Jarl (or something alike)?
Queen Potema, the "Wolf Queen of Solitude" (age 14)
Born in 3E 67, Potema was married to King Mantiarco at the age of 14 and took the title of Queen.
A year after the wedding of his 14-year-old granddaughter the Princess Potema to King Mantiarco of the Nordic kingdom of Solitude, the Emperor Uriel Septim II passed on. His son Pelagius Septim II was made emperor, and he faced a greatly depleted treasury, thanks to his father's poor management.
As the new Queen of Solitude, Potema faced opposition from the old Nordic houses, who viewed her as an outsider.
–The Wolf Queen, v2, first appearing in Morrowind
Emperor Reman (newborn, according to mythical account)
The mythical birth story of Reman said that as soon as he was born, he was placed on the throne as Emperor and spoke as an adult.
But after nine months that mound of mud became as a small mountain, and there were whispers among the shepherds and bulls. A small community of believers gathered around that growing hill during the days of its first churning, and they were the first to name it the Golden Hill, Sancre Tor. And it was the shepherdess Sed-Yenna who dared climb the hill when she heard his first cry, and at its peak she saw what it had yielded, an infant she named Reman, which is "Light of Man."
And in the child's forehead was the Chim-el Adabal, alive with the dragon-fires of yore and divine promise, and none dared obstruct Sed-Yenna when she climbed the steps of White-Gold Tower to place the babe Reman on his Throne, where he spoke as an adult, saying I AM CYRODIIL COME.
–Remanada, first appearing in Oblivion
There are two other people I found who are close, but don't really count as being in a governing position: