I don't think there's any reason to think that the Royal Navy ban on shanties was in force back in the days of Henry VIII. There wasn't really a Royal Navy as such anyway, just some ships belonging to the Crown.

I've sometimes wondered whether the raising of the Mary Rose back in 1982 might have been the inspiration for Stan Rogers' song about the Mary Ellen Carter, which Keith quoted in this thread's title - but I think the dates rule that out. (Though of course the project to raise her began some time before it was actually done.)