It is true that upon the death of Princess Charlotte of Wales (1796-1817) and her stillborn son, Queen Charlotte and King George had no legitimate living grandchildren nor great-grandchildren. This prompted a succession crisis. To prevent the throne passing to a distant relative, the King's unmarried sons were encouraged to look for wives and father children.
The house George gives to Charlotte is Buckingham House, originally built for the Duke of Buckingham in 1703. It was acquired by King George in 1761, the year of their marriage. The townhouse became known as Buckingham Palace after the addition of three wings around a central courtyard and it became the London residence of the British monarch on the accession of Queen Victoria in 1837.
The song playing as King George and Queen Charlotte ride in the carriage after the wedding is a classical rendition of "Halo," by Beyonce.
After Princess Charlotte dies Lady Whisledown refers to her as The People's Princess. Charlotte was never referred to as such. It however was a moniker that another Princess of Wales Princess Diana would be called in the press.