You should also notice the personal struggle that Wilberforce underwent, changing his beliefs. If you are trying to change someone it helps if you have changed your mind too.
In Amazing Grace it is shown in one Parliamentary House for dramatic effect but it really happened in two Houses, Commons and Lords. There were also committees, small groups that were even more important than the Houses.
Wilberforce also did not make personal enemies. Some of the MPs who voted for his bill in the end were violently opposed to it before. Wilberforce broadened his circle by embracing people who would disagree on everything except ending the slave trade.