There was no such title as the Duke of Chatsworth, nor was there a Duke or Duchess of Monmouth after the seventeenth century.
Neither Princess Feodora of Leiningen nor Princess Adelheid of Hohenlohe-Langenburg lived in England.
This episode which includes Prince Albert planning for the Crystal Palace first World's Fair held in 1851 would likely be taking place at least a year or two earlier in 1849-50. Lord Palmerston, portrayed by the excellent Laurence Fox looks like he's mid 30 years of age. The real Lord Palmerston dies in 1865 at age 80, which would make him 65 in this episode's time period.