Bill is first seen walking with a ladder. He was eventually asked by the Dodo and the White Rabbit to get rid of the monster inside the White Rabbit's house. It was actually Alice who had grown large because of a cookie that she ate, and Bill refuses at first. When Bill is forced down the chimney by the Dodo, the smoke causes Alice to sneeze sending him blasting off into the sky. What became of him afterwards is unknown.
Bill appears as one of Professor Ratigan's henchmen. He is first seen with the rest of the thugs and singing about Ratigan in the song "World's Greatest Criminal Mind". He then watches sadly as his friend Bartholomew gets fed to Ratigan's cat Felicia for calling Ratigan a rat. When Basil and his assistant Dr. Dawson show up at Ratigan's lair, Bill along with the other thugs laugh at Basil and tie him and Dawson up. When the thugs tie up the real Mouse Queen and use the robotic queen to fool the crowd, Bill gives notes to Flaversham on what to say into the speaker of the robot mouse. When Basil, Dawson and Olivia show up at the palace, they tie up Bill and the other thugs and the Queen has them arrested.
Bill also appears as one of Professor Ratigan's henchmen in The Great Mouse Detective. He is never called Bill in the movie, but his appearance and clothes (and Cockney accent) are identical to that of Bill in Alice in Wonderland. The only difference in his appearance is that he does not wear his black coat in The Great Mouse Detective, whereas in Alice in Wonderland, he does.
We don't know how a character from Wonderland can be part of Ratigan's band in London. Perhaps when he was ejected out the chimney of the White Rabbit's house, he flew so high he went out of Wonderland (which is underground) and ended up in England, where he became a bandit.
It is also possible he fell into Wonderland sometime after Ratigan's defeat.
It should be noted the two films take place during Victorian England, and thus the same time period.