Hugill mentions two shanties as of probably Elizabethan origin, with one even earlier - "A-Rovin'" being the Elizabethan, and "Haul on the Bowline" being of probable earlier genesis. Admittedly, the reasoning given for the dating is that the bowline ceased being a 'heavy work' line some time in apparently the 14th century, and thus not needing a full shanty for work purposes.
'Shanties of the Seven Seas' covers a lot of this topic, along with Doerflinger's work and a few others... After reading what they've written, as well as other independent sources (encyclopaedias, national histories, etc.), I don't agree that African influences were responsible for the evolution of the sea shanty. They were a definite shaping influence, but not the MAJOR influence.
The closest conglomerate explanation that I'd agree to is that sailors were using 'call outs' or 'sing outs' (rythmic yells and stamping in time to the work) to time the pulling. While African tribes may have been doing this before sailors sailed, it isn't much of a stretch to imagine that sailors could independently develop similar ideas due to the nature of the work (i.e., 'if we don't all pull together, that sheet ain't going nowhere, so everyone pull on my mark and signal that you're doing it by responding with a yell'). From there, it's even less of a stretch to imagine the sailors, once presented with a steady beat, would start using favorite songs from home to time the work with, and then from there start developing songs of their own. Art finds the strangest beds to grow from...
In all this, it's entirely reasonable to assume that EVERY nation that sailors came in contact with would add to the blend. After all, it was a common practice to bolster diminishing crews by taking (hopefully) willing natives to sea as pilots, and some would stay on as full time sailors... Some ships would even man whole crews of 'natives', as noted by Dana in 'Two Years Before the Mast'. There, in California, he records a ship entirely manned by native "Kanakas" (Hawaiians).
Plus, as sailors signed on to different routes, their experiences would go with them... so on an Atlantic trade vessel, you might wind up with a Pacific South Seas sailor sharing the songs and shanties he'd learned on his previous voyages... and vice versa. That's part of what makes tracing where shanty influences definitively came from so difficult.
Anyway, that's my take on it. Do with it as you will...