Two hangings are shown in the film, in the style of a "long drop" designed to break the neck and cause instant death. This style was first introduced in the 1870's by William Marwood as a more humane approach. Before then, and certainly in 1828-1829 when the film was set, the victim was simply suspended by a rope around the neck, and choked to death by suffocation.
Hare comments that his wife Lucky has "fallen off the wagon" when she resumes drinking alcohol after a period of abstinence. The idiom of "falling off the wagon" would not be in used in Scotland at this time. The phrase originated in America very close to the start of the 20th century.
When Burke and Hare stop for a drink and Burke says that they have no money, Hare plucks a half-crown from underneath Burke's cap. The half-crown that is used in the film is from 1902 - 1910 which is nearly 100 years after the film is set.
In the "multiple dissection" scene Tom Wilkinson uncovers each of the four dead bodies one at a time. But, following a quick edit to the applauding students, in a closer shot one of the bodies is covered up again.