It's generally agreed that chanteys were not routinely permitted in the Royal Navy of the "Great Age of Chanteying," ca.1830-1880.

But there's at least one apparent reference to RN sailors of the previous generation singing while hauling.

From Marine Lieut. Henry Barnet Gascoigne, "Gascoigne's Path to Naval Fame," 2nd edition (1825), p.75:

"Beating to Windward...
Now with a song the Bow-lines well they Haul;
The Weather Braces then Haul-taught of all."

On the other hand, as Hugill and others observe, the bowline (regardless of the well-known chantey) was not the sort of rope that needed a whole song. Gascoigne himself defines it as follows (p.167):

"Small ropes attached the Leeches or sides of the sails, to bowse or draw the weather side forward when sailing upon a wind, that is, beating to windward."

But possibly what Gascoigne heard was merely a "sing-out" or a single line.

According to correspondent in the Times (Aug. 26, 1918), Gascoigne served in the RN frigate Melpomene in 1805.