I've not seen this series. However, regarding Prince Arthur's cause of death, here's what the Wikipedia article about him states:

In March 1502, Arthur and Catherine were afflicted by an unknown illness, "a malign vapour which proceeded from the air." It has been suggested that this illness was the mysterious English sweating sickness, tuberculosis ("consumption"), plague or influenza. While Catherine recovered, Arthur died on 2 April 1502 at Ludlow, six months short of his sixteenth birthday.

Regarding sweating sickness, following is some of what Wikipedia's "Sweating sickness" article explains about that particular illness/condition:

Sweating sickness, also known as the sweats, English sweating sickness, English sweat or sudor anglicus in Latin, was a mysterious and contagious disease that struck England and later continental Europe in a series of epidemics beginning in 1485. The last outbreak occurred in 1551, after which the disease apparently vanished. The onset of symptoms was sudden, with death often occurring within hours. Sweating sickness epidemics were unique compared to other disease outbreaks of the time: whereas other epidemics were typically urban and long-lasting, cases of sweating sickness spiked and receded very quickly, and heavily affected rural populations. Its cause remains unknown, although it has been suggested that an unknown species of hantavirus was responsible.