Prior to the Tudors came the Plantagenets, after them were the Stuarts. The Hanoverians were the rulers (Electors) of Hannover and named after the house of Hannover, the equivalent of their last name. Before the Plantagenets Kings didn't really have last names - surnames evolved some time in the middle ages.
Look at coins from the respective reigns. Those of the James's bear the legend Jacobus Rex whilst those of the Charles's, Carolus Rex. Before the abolition of pre - decimal currency, and even afterwards whist silver coins of the previous reigns remained valid, the legend on the obverse of coins of the reign of George VI carried the legend, Georgius VI Rex.