While it would certainly not be the most secure circumstance to enter into a war with. it is absolutely possible for a soldier to survive one armed with only a longbow. After all a weapon is a weapon and depending on the skill of the user with his weapon of choice, he may have a good chance of survival.
If you are really good at killing people with a long bow then you stand a pretty good chance. Granted, you do leave yourself more open than somebody with a rifle but stranger things have happened. If you are quick and deadly it really does not matter what kind of weapon you have.
In any war, the scales are tipped based on advantages, but not all advantages are based on which weaponry is used. A soldier's instinct, intelligence, training and capability are his greatest advantages, not the weapon chosen. Soldiers have been known to survive for no reasonable reason in the harshest circumstances. Many say it is fate; others say it is faith. Either way, the weapon used is not the only factor involved in winning a war or surviving.
No, of course it would be impossible for a WWII soldier to have fought using only a longbow. This was a war of heavy weaponry, advanced machine gun technology, versatile rifles, and widespread handguns to back up any primary weapon. A soldier with a longbow, even though it would be possible for this soldier to earn a kill or two, would not be able to survive for very long, as the time it takes to fire and reload a longbow, as well as the difficulty in aiming relative to a gun, would make it very easy for this soldier to be shot and killed.