"If by British Navy you are referring to the Merchant Service then singing of shanties was not banned.
If you are referring to the Royal Navy then there was never any need to sing shanties to work the ship as all work was undertaken to direct order or by pipes on a bosun's call that would always be heard in action or in a storm whereas a voice would not."
Interesting implication that Royal Navy ships sailed in worse storms than merchant navy ships. Maybe there was a difference in practice, but if so it was because of a difference in attitudes to the workforce and discipline etc. The "Royal Navy ships were better manned" argument scarcely holds water either, unless one assumes that prize crews, or decimated crews that had survived battles & scurvy, were allowed to sing shanties.
Not that it matters in respect of folk clubs where, inevitably, it is far more common to hear pshanties, i.e. barber-shop shanties, rather than work songs.