If you assume that none of the first 5 cards (to be discarded) belong to the Hearts Royal Flush (10-J-Q-K-A), then the odds of hitting a Heart Royal in sequence from a 47-card remaining deck is: 47 x 46 x 45 x 44 x 43 = 184,072,680.
Divide this by 200 hands per hour = 920,363.4 hours.
Divide this by 8,760 hours per year (no sleep / no breaks ... after all, you're dead!) = 105 years.

To calculate the number of hands spoiled at the outset because a Heart 10-J-Q-K or A comes up, that's 5 cards you have to dodge out of 52, or about 10% of the deck (rounded). So, add an extra 10 years to the task for a grand total of prox. 115 years to accomplish.