English longbowmen served a number of purposes when they were either out of ammunition or couldn't effectively shoot anymore. The main one is that they took up hand weapons and joined the fight in support of the main force of men-at-arms. They often attacked as a flanking force and swarmed isolated groups of enemy troops. Longbowmen would also assist behind the lines with managing prisoners (either keeping them under control or executing them) and sometimes finishing off the enemy wounded. Their utility even outside of their main function as bowmen was one of the reasons for the increased importance of archers in English armies throughout the 14th century.

Oddly enough, even though most vase paintings show them with just one or two javelins, we never hear of peltasts running out of missiles in Classical Greek sources. In the prolonged running battles that were the peltasts' most famous victories, they are described as casting endless volleys of javelins at their enemies from a seemingly inexhaustible supply.

The probable answer to the riddle is that peltasts actually did not throw their javelins en masse. Their "volleys" involved the missiles of only a handful of men each time, slowly sapping the enemy's numbers, energy and morale. Since light troops always had the advantage of mobility, it was in their interest to prolong the engagement and exhaust their heavily-armed opponents, rather than to try to get maximum impact with a single huge volley. They therefore tried their best never to run out of ammo. During the March of the Ten Thousand, peltasts even picked up the massive longbow arrows of a local tribe and used them as replacement javelins:

They had bows nearly three cubits [about 5 ft] long and their arrows were more than two cubits, and when they shot, they would draw their strings by pressing with the left foot against the lower end of the bow; and their arrows would go straight through shields and breastplates. Whenever they got hold of them, the Greeks would use these arrows as javelins, fitting them with throwing loops.

-- Xenophon, Anabasis 4.2.28

If they did run out, though, they could still serve as light infantry, armed with slashing swords, thrusting spears and light shields. They were well suited to pursuing heavy infantry, which was slowed down by its equipment. Fresh light infantry was a prized asset for when the enemy routed.

As for archers, in Classical Greece we have a few examples of them running out of arrows. In one case, again during the March of the Ten Thousand, they simply picked up the arrows the enemy was shooting at them and continued their work:

The Persian bows are also large, and consequently the Cretans could make good use of all the arrows that fell into their hands; in fact, they were continually using the enemy's arrows, and practised themselves in long-range work by shooting them into the air.

-- Xenophon, Anabasis 3.4.17

In another, the depletion of their ammo stores meant the end of their usefulness - and the end for the hoplites they supported:

As long as their archers had arrows left and were able to use them, they held out, the light-armed Aitolians retiring before the arrows; but after the commander of the archers had been killed and his men scattered, the soldiers, wearied out with the constant repetition of the same exertions and hard pressed by the Aitolians with their javelins, at last turned and fled, and falling into pathless gullies and places that they were unacquainted with, thus perished.

-- Thucydides 3.98.1-2