They had built a missile inside the column and planned to detonate it in 2017, destroying the Earth and distracting the Charnal horde, with whom they were at war. To protect this operation, they had several stone lions, hidden by perception filters, patrol the area. The Doctor managed to appeal to Peto and Grissell and they broke free.
The Doctor spent two weeks lowering the missile into a shaft beneath column and building a floor to cover it. By this time it was November, and the statue of Nelson was being added when the Doctor departed. (PROSE: The Lions of Trafalgar)
In a timeline where the Daleks exterminated humanity in 1963, Nelson's Column, along with London and presumably the rest of the world, was in ruins. When the Eleventh Doctor and Amy Pond arrived in the Square in 1963, the Doctor's TARDIS landed next to the statue's toppled head. The timeline was restored when the TARDIS crew travelled to Skaro to destroy the Eye of Time. (GAME: City of the Daleks)
The Seventh Doctor materialised his TARDIS around Nelson's Column in 1976. The Monk caused the Doctor's TARDIS to change its appearance from that of the column to that of the Statue of Liberty. (PROSE: No Future)
The Eleventh Doctor visited Nelson's Column with Amy and Rory in 2012. He stated that one of the four plinths was empty because there had not been enough money to put a statue on it. Amy believed that they had been showing works of art on the empty plinth and changing them out every year or two. The Doctor believed they were looking for a permanent piece of art for the space. (PROSE: Magic of the Angels)
In an alternate timeline, Nelson's Column was destroyed by the Daleks when they invaded Earth in 1903. After the invasion was defeated, it was replaced by the British Empire with a statue of the Sixth Doctor, who had aided in their victory. (AUDIO: Jubilee)