I don't know much about the history of shanties, apart from what I've read in books about folk music such as Lloyd and Hugill, but as a general point, please beware on what constitutes evidence. Curmudgeon lists an impressive array of references, but they are all secondary sources. In such cases, it's not uncommon for a sort of folk- process to occur; one writer prints a statement, another copies it, someone else reads the same thing twice in different books, so it MUST be true, and perhaps adds a bit extra by way of comment, which is widely read, and thereafter no one ever checks up raw sources because it's such a widespread bit of information.

A classic example of this (not trying to drift the thread) is a common reference in canal books and guids to Betton Wood on the Shropshire Canal. It's often stated that this is (or was) haunted by a screaming ghost, and that the old boatmen would not willingly moor there. This intrigued me, as I had read M.R.James's ghost story, "A Neighbour's Landmark", in which a Betton Wood is haunted, so I asked around a little. And the trail seems to lead back to L.T.C.Rolt, a pioneer of recreational canal boating, who wrote in his classic book, Narrow Boat, that he moored near the wood in the late 1930s and was reminded of the story by the name and gloomy location.

So I don't know if shanties were really forbidden in the Navy, but to find out for sure you'd have to study diaries, logs, eyewitness accounts etc. from the relevant periods.