They show Elizabeth of York's aunt Katherine who married Jasper Tudor as a woman in her 50s or 60s. She was in fact about the same age as Henry VII and was in her mid to late 30s when the series takes place.
Throughout the series, the characters speak using contractions. Contractions weren't common in the English language until the 1700s.
There is no evidence that Henry or Margaret Beaufort are responsible for the deaths of Elizabeth's younger brothers, Edward V and Richard (known today as The Princes in the Tower) .
The shows Margaret Beaufort and the dowager Queen Elizabeth hating each other and Elizabeth resenting Lizzie marrying Henry. In reality, both mothers agreed uniting the Houses of York and Lancaster would end the War of the Roses and cement Henry's claim to the throne.
The same two boys play Arthur and Henry for the last couple episodes. However the boys were young. By the ending of the series Arthur and Henry would have been about 13 and 10 not 10 and 6.
Church bells are heard ringing changes, but full-circle change ringing was not developed until the 17th century. In Tudor times the bells would only have been rung by chiming, which sounds very different.