The derisive connotation of Uncle Tom came from later works that were based on or derived from Harriet Beacher Stowe's book, Uncle Tom's cabin, which was released in 1852. Copyright law back then did not give authors control over characters created in a novel, so Harriet Beacher Stowe had no involvement or control over these later characters, or the use of the name Uncle Tom.
There was an entire genre of minstrel shows, usually performed by white men in blackface, that were used to make fun of African Americans and tended to be pro-slavery. The main character in these shows was Uncle Tom, and the shows were known as Tom Shows. Unlike the character in Harriet Beacher Stowe's book, the Uncle Tom in these shows conformed to the cultural stereotypes of the day. Uncle Tom was usually portrayed as dumb, old, balding, and a target for ridicule. This changed the cultural understanding of the term Uncle Tom from that created by Harriet Beacher Stowe, to that created by the numerous minstrel shows that portrayed Uncle Tom as a fool.
These shows continued in the United States for the better part of a century, and were probably much more widely known than Harriet Beacher Stowe's book. This resulted in a very different understanding of the term Uncle Tom from what was was originally intended.