As a longbow archer, I can tell you that there will be even more problems.
Instinctive archery requires that you have shot so many times you know the arrow's tragectory intuitively. The wood arrows tend to soak up water and become heavier, htereby altering the tragectory. Further, the flethching, depending on the type of feather, when it gets wet becomes effected. THe modern standard for fletching is turkey feathers, and they tend to soak up water, lose their shape, as it were (It looks like they get crushed), and makes the arrows heavier. I believe in older times, goose feathers were used, and goose feathers, although not as durable as turkey feathers were much better at repelling water due to oils in htem. The feathers were glued on and wrapped on with some kind of thread, so if htey did get wet, the fletching would still stay on the arrow.
For the strings, today we use Dacron and a material called fast flite, back then, I've heard of linen strings being used, as well as sinew.
What you've got to remember is that the stirng when strung is always under some sort of tension. War bows back then were under a lot more tension, and consequently the strings would stretch over time, each string being good for a short period of time, after which it having stretched so much, you'd put a new string on (I suppose they used the old strings for other things, htey were still good for repairs, but not for bows.)
Even today, a modern string is only good for so many shots (I make my own and can tell you).
Here's the deal though, even today, each bowstring is waxed, so they will, to an extent, repell water, but as the organic material absorbs water, I suppose it will lose some of its structure and stretch, thereby, vompletelythrowing of the tragectory of the arrow.
THere is now also a rest on the bow, made up of fur or feather to let the arrow slide over and reduce the friction. As water is absorbed by this, it also reduces the speed of the arrow, increasing friction.
Also, the wood in the bow.
There are a couple things going on, the bow wood has a cellular structure that gives it its rigidity and flexibility. (Some woods are more flexible han others, naturally) By having water soak into the wood, the limbs ofthe bow gain weight, slowing the bow down. Further, the water destroys the cellular structure of the wood and also the elasticity, thereby slowing the bow down.
In sum, the arrow will fly at a significantly slower speed, throwing off the archer's aim, lessening the force of impact, and if the water continues, destroying the arrow and the bow.