When Bligh, Fryer and Christian are in Bligh's home planning the voyage, Bligh refers to a route that would take them around the coast of 'Australia'. But at the time of the Bounty's voyage in 1789 what we now know as Australia was instead universally called New Holland - a name which also appears on Bligh's map and which he later uses after being cast adrift. 'Australia' only came into common usage in the early 19th century; it gained official status in 1824.
The Tahitian chief and his daughter are shown wearing feathered headgear. The red/orange tipped feathers are from "Lady Amherst" pheasants. These are native to the remote mountains of SW China. None were brought out of that area until the 1820s. The mutiny happened in 1789.
The continent of Australia is referred to as both "Australia" and "New Holland". The latter is more correct for the time - the name Australia would not be adopted until the 1800s, although it may have sometimes been referred to as "Terra Australis".
Among the midshipmen on the historical Bounty voyage were a Thomas Hayward and a Peter Heywood. These two are understandably combined in the film into a single character. It was, for example, Mr. Hayward who fell asleep on watch the night three men deserted ship, but Mr. Haywood who stayed behind when the Bounty returned to Tahiti. While this composite character is named "Thomas Heywood" in the credits, Fletcher Christian calls him "Peter" near the end of the film.
When the three deserters are being flogged, many of the Tahitians are shown harming themselves in protest, including Christian's consort Mauatua. However in the next shot she shows no visible scars from this.
When the Bounty left Tahiti before the mutiny it actually sailed west. Cape Horn was never an option on the return journey as the breadfruit would not have survived the cold weather. So this was certainly not a factor in the mutiny.
When Bligh and his launch arrive in Coupang, the launch survivors are depicted as starving and near death, seemingly to imply that they had been constantly at sea since the Mutiny on the Bounty. Historically, Bligh's launch had reached the eastern coast of Australia two weeks earlier and had repaired their boat, stocked up on food and water, and had only then set to sea again to reach the Dutch East Indies. Thus, when they reached Coupang two weeks later, the launch and her crew were in fairly good shape.
At the beginning of a meal, the officers toast the King, but do so while sitting. British serving officers always stood while toasting the monarch, even though only in the captain's cabin of a large ship would there be enough headroom to stand upright. When William IV, a naval officer since he was about 13, became King in 1830, he allowed serving officers to toast him while sitting, but this was more than 30 years too late for the officers of the Bounty.