The verse:

"The Vicar he stood at the alter
And said who gives this woman away
A bloke off the "Hood"
Said blimey I would
But let every dog have it's day"

That would date the song after May 1920, when the year HMS Hood was commissioned from a new build into the fleet, and May 1941 when it was sunk in action against the Bismark in the Denmark Strait.

Always known it as a "Standard" Sods Opera song. No Idea who actually came out with it but it must have been written after the outbreak of the Second World War (1st September 1939 and 24th May 1941). In 1939 HMS Hood was in Portsmouth and the WRNS having been formed in 1917 and disbanded in 1919 were only reformed at the outbreak of war in 1939 so there could have been no Jenny Wren brides between May 1920 and August 1939.