As part of her audition process Nikolaj Arcel took Alicia Vikander out drinking to see how well she could understand Danish. Vikander faked that she could understand and speak it well but at the end of the process when Arcel told her she had the role she didn't understand what he was saying until he switched to Swedish.
In 1769, when Queen Caroline Mathilde is taking tea with the Dowager Queen Juliane Marie, they are clearly seen to be drinking from Danish Flora Danica cups. The Royal Danish Porcelain Factory was not established until 1775 and was the service was not put to use by the Danish Royal Family until 1803.
While all of the characters all speak Danish in the film, the court language in Denmark at the time was actually German. In real life neither Count von Bernstorff nor Johann Struensee spoke any Danish, and it is more than likely that Christian and Caroline also conversed in German rather than the "people's language."
This movie is based on historical events. They were chronicled in the critically-acclaimed 2002 novel, The Royal Physician's Visit, by Swedish author Per Olov Enqvist.