It would be uncommon from what I have seen for a ship on active war duty to be hauling freight or anykind. By Henry VIII time you have a much more defined system of class; warships, transport, even scouting ships. Anything on the mary Rose when it went down would have had to be ready for use. If I remember right an article that just came out on the mary rose covers this as well but not I cannot remember where it was but it staves would only have been in the last 2-3 months past it came out (I read it when waiting for a meeting to start...).

Earlier it is possible as the ships were often just lumped together and the number of archers and men at arms on the ship made any merchant vessel a warship. The scale of bows and arrows made in england is massive. I know that the tower has tens of thousands of final product bows made during prep for war. One fletcher from london when he died left half a million arrows in his shop..... ready for the continuation of the Hundred Years War. I cannot even imagine the huge scale these were being made sometimes. Southampton has men entering the port and gate with thousands of bowstaves and bows. Totally incredible organisation, London even more so.

RPM

Glennan is touching on an important point about longbows and arrows; you need tons of them for a campaign. Sure the recurve is a better bow but it takes longer to make. When the English went to war they took thousands of archers with them and bow break so you need extra bows too. The simple design of the longbow made it easy to mass produce.