I've gone over those called by Joseph Smith here, here and here. I would argue that in some cases, Joseph was naive with some of the men he put in important positions, and he saw some of them turn into bitter enemies. Brigham Young's callings went out to those who'd shown a little more steadiness, men who'd gone through a little more of the refiner's fire. Also, he made sure each one was okay with polygamy.
So if Joseph's main call was to restore the Gospel, and Brigham's main call was to organize the church and ensure its survival, Taylor's call was to transition the church to have the institution as Christ's church be more important than any one man. Joseph and Brigham both had cults of personality, which allowed them to be beloved by some, reviled by others. Taylor brought a real kindness and diplomacy to his time as leader of the church, even though persecution due to polygamy increased greatly during his tenure.
So as far as restocking the Quorum of the Twelve, how did he do?
Called by John Taylor to be Apostles:
Moses Thatcher, Francis M. Lyman,
John Henry Smith, George Teasdale,
Heber J. Grant, John W. Taylor
(February 2, 1842 - August 21, 1909)
Apostle - April 9, 1879
Dropped from Q12 - April 6, 1896
It was explained that John Willard Young and Daniel H. Wells did not join the Quorum of the Twelve upon Brigham Young's death due to an excess of Apostles, but after Orson Hyde died, John Taylor called Thatcher. Taylor didn't reorganize the First Presidency until 1880, so Thatcher became #12 while Young and Wells were still outside.
Thatcher's parents joined the church when he was one year old. He grew up mostly in California, but his family moved back to Utah in 1857 when Brigham was calling most of the California Saints to move back to Utah. Thatcher served two different missions for the church and later served as a mission president in Mexico.
He was called to be an Apostle at age 37, but in 1896, when Wilford Woodruff was president, Woodruff issued a new policy that stated LDS church officers had to get approval from their church leaders before running for or holding political office. The other Apostles signed on to it, but Thatcher refused, believing it went against the church's policy of political neutrality. He was dropped from the Quorum of the Twelve, but he retained the office of Apostle and remained active and in good standing with the church for the remainder of his days.
(January 12, 1840 - November 18, 1916)
Apostle - October 27, 1880
President of Q12 - October 6, 1903
Francis was the oldest son of Amasa Lyman, who became an Apostle when Francis was two years old. When he was 11, he moved with his dad and Charles C. Rich to settle San Bernadino, CA. he was called on a mission to Great Britain in 1857, but it was delayed so he could help move Saints from California to Utah during the Utah War. After he got married to Australia native Rhoda Taylor and had a child, he left for his mission in 1860.
When he returned, he was directed by Brigham Young to settle Fillmore, Utah. He was serving in Fillmore's High Council when his father was excommunicated. Francis took on a second wife in 1869, George A. Smith's niece Clara Callister. He would eventually marry Clara's sister Sarah as well.
He served a European mission in the 1870's, proselyting in the United Kingom, France, Germany, Switzerland and Denmark. Upon his return home he was called to move to Tooele and preside over its stake. He also got into politics as a member of the People's Party, while the county was dominated by the Liberal Party. In the county races of 1878, the Liberal Party refused to count the votes and declared themselves the winner, but Francis headed up pushing the matter through state and district court, and the People's Party was declared the winner in 1879.
When John Taylor reorganized the First Presidency in 1880, he called Francis (age 50) and John Henry Smith (age 42) as apostles. Francis was older and therefore took his place in senority line.
He spent his early years as an Apostle proselyting among the Native American tribes of the West. He also served as president of the European mission. He was President of the Quorum of the Twelve in 1903 under Pres. Joseph F. Smith and was therefore next in line to be prophet until he died of pneumonia at age 76.
JOHN HENRY SMITH
(September 18, 1848 - October 13, 1911)
Apostle - October 27, 1880
Second Counselor - April 7, 1911
John was the son of apostle George A. Smith and the father of future prophet George Albert Smith. He served his mission in England, and he had nineteen children by two wives. He was called to be an Apostle in 1880.
He took over as president of the European mission in 1882 and was the one who discovered the infidelities that eventually got fellow apostle Albert Carrington excommunicated.
He was a Republican and active in politics, and he was a key figure in helping Utah become a state in 1895. He and George Albert were the only father-son combination in church history to serve in the Quorum of the Twelve at the same time. John became second counselor in Joseph F. Smith's presidency after John R. Winder died, and he served there until he died from a pulmonary hemorrhage at age 63.
(December 8, 1831 - June 9, 1907)
Apostle - October 16, 1882
When Lyman and Smith were called to be Apostles in 1880, there were still only 11 members in the Q12. After Orson Pratt died in 1881 (the last original member of the Q12 called by Joseph Smith), there were only 10 members, and a few months later, Teasdale and Heber J. Grant were called.
Teasdale, like Taylor, was born and raised in England. He joined the church at the age of 20 and served in many church leadership positions there until he moved to Utah in 1861. He would later go back to England and other countries in Europe as a missionary. He also served in the Southern US mission. By profession he was a teacher, so between missions he'd find employment in different school districts.
He was 50 when he was called to be an Apostle, and soon he'd preside over the Mexico mission. He died at age 75 from an "intestinal obstruction."
(November 22, 1856 - May 14, 1945)
Apostle - October 16, 1882
President of Q12 - November 18, 1916
Prophet/President - November 23, 1918
Heber was only nine days old when his father Jedidiah, 2nd counselor in the First Presidency, died. His mother Rachel Ivins remarried to Jed's brother George, but he was an alcoholic so she left him and raised Heber on her own. She refused to take church assistance and made money from renting rooms in their house. Growing up poor, Heber was determined to be successful in business.
He married lifelong friend Lucy Stringham in 1877. It was a long courtship, as he'd been dating Emily Wells (daughter of Daniel H. Wells) for a time, and Lucy wanted to make sure she wasn't just a replacement for Emily. Heber married Emily in 1884, as well as Augusta Winters. he was the last prophet of the LDS church to practice polygamy, but by the time he became prophet in 1918, only one of his wives was still alive.
Heber started many businesses before and after becoming an Apostle at age 25. He used his business acumen to help get church finances under control.
He organized a mission in Japan in 1901 and served as its first mission president. He was immediately called in 1903 to serve as mission president of the European mission, and then the British mission until 1905.
When he became prophet and President of the LDS church at age 62, he was the first one to have been born after the death of Joseph Smith. He was instrumental in the building and dedication of temples outside of Utah, and in making sure the last of the polygamy holdouts gave up the practice. When he excommunicated a stake in Arizona in 1935 for refusing to give up polygamy, it led to the formation of what would become the FLDS church.
He was a Democrat, but often disagreed with FDR and he fought against the repeal of Prohibition. He was mortified when Utah agreed to end Prohibition. His service as prophet was the second-longest reign in church history, behind only Brigham Young.
JOHN W. TAYLOR
(May 15, 1858 - October 10, 1916)
Apostle - May 15, 1884
Removed from Q12 - April 1905
Excommunicated - 1911
Taylor lived most of his life in Idaho and had six wives. After Charles C. Rich died, Pres. John Taylor called his own son to fill the vacancy in the Q12. John W. was ordained on his 26th birthday. As an Apostle he spent a lot of time helping build the church in Canada.
After Pres. Woodruff issued the 1890 Manifesto, John W. continued to take wives in secret. He and Matthias F. Cowley (who became an Apostle in 1896) were removed (or resigned) from the Q12 in 1905 over their refusal to comply with the end of polygamy. He was eventually excommunicated and died of stomach cancer at age 58. His Apostleship was posthumously restored in 1965.