I've used sinewed wood bows and composites here in the England many times in the rain with no ill effects at all. I shot the 3D nationals a few years ago and it poured down pretty much the whole time, both days. 80 r.h., with a composite horn bow. It showed no loss of reflex
No finish is totally waterproof but some are much better than others.
For wood/sinew bows I use tung oil. 5 coats total with the first coat cut 50/50 with white spirit and 2nd coat 75% tung oil/25% white spirit. Tung oil brings out the grain nicely, is very easy to apply and is a very water resistant finish. Only use pure tung oil with no drying agents added for best durability. It is very easy to re-apply and hide any scratches that happen over the years.
I've tried just about every finish over the years and tung oil is a strong favourite of mine