[Jim McDougall]

This is truly an interesting topic, and as I'm sure most are aware this is pretty much a waltz through a bureaucratic administrative web which seems to have begun around early 1300s.
It sounds strange but the keepers of the 'Privy Wardrobe' were the first official 'keepers of ordnance'.
This strangely appointed office for royal stores became more clearly designated to ordnance in the time of Henry VIII in the 1540s.
In 1683 Charles II issued a warrant specifying use of the 'pheon' on official stores, which we may presume included arms.
In 1806 the board was ordered to mark ordnance stores again.
This chronology is admittedly vague, but it seems such is the history of the Board of Ordnance.

Most items of issue from the Napoleonic period until 1855 were marked BO topped by the arrow.
After 1855, it was changed to WD under the arrow (War Department).

So the question is:
When was the broad arrow first implemented as a marking on arms?

As Will has shown, we know artillery had it in 1760, so we can presume that larger segment of ordnance had such use in place then and earlier.

This background is not the answer needed, just some perspective, pending search further re: actual arms markings.

Attached is a post 1855 wooden canteen with the WD instead of BO accompanying the broad arrow.

From: "Treasures of the Tower of London"(1982, A.V.B. Norman, G. Wilson)
While swords remained in the control of regimental colonels until c. 1788, firearms were purchased and controlled by the Tower and Board of Ordnance.
It would seem that the locks of guns were marked with the crowned broad arrow as early as 1731, probably earlier, from these examples;

#97, Long Land pattern musket , crowned arrow, date 1731
#98 cavalry pistol date 1744 crowned broad arrow
#99 cavalry carbine c.1770 crowned broad arrow

I had hoped that bayonets might yield some clues, so to "The Plug Bayonet" (R.D.C.Evans, 2002);
It would appear that the Board of Ordnance was involved in degree with bayonet supply (then plug bayonets) by c. 1685, however the Board preferred that colonels privately purchase these. Most of the bayonets seen have the dagger mark of the LCC (=London Cutlers Co.) but obviously no crowned arrow.

Possibly in Queen Anne period (1702-14) the advent of the socket bayonet may have changed that, but remains to be seen.

On an aside, the East India Company, which clearly had its own bale mark, seems to have pretty faithfully marked all their gun locks, and the socket bayonets marked accordingly. With swords, as I was told by David Harding ("Small Arms of the East India Company") the Company did not place bale marks on its swords.
Perhaps this may suggest BO arrow markings might be on socket bayonets for British government ordnance?