The name does appear in the list of Strangers in the metropolis for London during the reign of Elizabeth I
1559, 1576, 1582, 1594 & 1600 Peter Tryon, alias Trewel, of Flanders
Probably this same man noted also in 1571: "Peter Trewe(l) and Marie his wife of Antwerp, Merchant and Denison having been in this ward for years, and has 3 children and a man servant."
1582, 1583, 1597 Garret Tryon, a Dutch Silk Twister
1600 Moises Tryon
After 1556 Phillip II became ruler in the Netherlands, and as a zealous Catholic, great persecution began there against the Protestants, under the Spanish inquistion, which eventually ended up in all out revolt, and after decades of struggle, the independance of the northern states, the basis of modern day Holland. So given the timing the Tryons may well have been amongst the religious refugees fleeing this persecution, and could have used the Dutch church in london which was opened by them in the 1550's. That said economic migrants had been coming from this area of the Netherlands for at least two hundred years before this, though in much smaller numbers.