Well, here's some more thoughts about the timing of the growth of the British merchant fleet. I don't think you can really blame it on the shanties, though.
British sea power was growing during the C16th but started to become particularly important at the end of the C16th after the defeat and destruction of the Spanish Armada. There was a large increase in traffic to the Americas at the end of the C18th. This was because of the Napoleonic war when, for the first time, the French began to compete seriously with Britain for trade with "The Americas" (by then Britain had a virtual monopoly in Europe due - I think - to Spain's involvement in the War of the Austrian Succession). By then, also, the British were heavily involved in the trading of slaves from the West African coast to "The Americas".
It would appear to be the case that, by the end of the Elizabethan era, (about 1600) traditional shanties, such as "A-Rovin", were already in common use - probably in The Netherlands, Flanders and France also.
PS ... re: the spelling ... it obviously has an important influence on how you understand the etymology of the word and hence what light can be put on the origin of the songs. Whall, in the introduction to his 6th edition (1927) is with Jody (historically rather than etymologically) to some extent in that in that he claims the earliest collections of Shanties were called "Songs from the Shanties" and later "Shanty songs" before being called simply "Shanties".