This is, somewhat strangely, not discussed in contemporary law codes. A later Continental code (11th century) does say that a warrior's weapons should not be looted, and Michael Swanton thought that a similar prohibition against corpse robbing probably existed in England. There's no evidence that it did, however, so we're forced to look at more indirect evidence to answer your question.

First, who owned weapons? Cnut passed a law in the early 11th century that warriors who died had to give (return?) a full set of war gear to the king. Cnut was asserting his role in the inheritance of weaponry -- and the social status that went along with bearing arms. Before this law, however, it's much more difficult to determine who owned weapons and how they passed between hands at a warrior's death.

The poem Beowulf (written possibly in the early 8th century, but no later than the early 11th) describes kings giving weapons to warriors (like the hero Beowulf) who then give these weapons to their own lords. Beowulf, it seems, cannot keep the gifts he personally received himself: they have to go to his own master first, and his master then decides whether he will give them back. However, even as Beowulf is giving the weapons he received to his king, Ecgtheow, he is telling a story, and in this story a man walks into a hall wearing a sword that he looted from a corpse, and this stolen weapon causes all kinds of trouble when it's recognized by the family of the man from whose body it was taken. What should we conclude? Perhaps that, when the poem was written, there was an idea (supported by the poet him/herself) that weapons should be controlled by lords, but a reality that was much messier. Grendel himself (the monster Beowulf fights) is described as weaponry (his skin is armor, his fingernails are spears), that is: Hrothgar's big problem is weapons that walk into his hall when they aren't wanted. In a society where weapons were widespread and political power was diffuse, controlling access to and ownership of weaponry must have been nigh impossible.

We have little archaeological evidence of what happened after a battle, with a few exceptions. Most of our cemeteries from this period are community graveyards. They sometimes contain bodies that were killed in a fight, but buried carefully by loved ones -- ie, bodies that were brought back from battle. We don't have an Anglo-Saxon battlefield that we can excavate to see what was and wasn't taken.

We do, however, have the Staffordshire Hoard, a collection of thousands of fragments of mostly gold discovered by a metal detectorist a few years back. This hoard is a collection of gold that was ripped off of other things, mostly swords: it contains pieces of something approaching 100 different weapons. It also has most of the gold foil from a helmet, and a golden cross ripped from ?book, ?reliquary, or other fancy church-related object. This material was clearly collected because of its bullion value: there are lots of pieces of beautiful objects, hacked apart to get as much gold as quickly as possible. There aren't, significantly, any whole weapons in the hoard. As far as we can tell (perhaps this will change as analysis continues?), the sword blades weren't kept by this collector, just the precious metals. The objects in the hoard date it to around the 8th century.

Who hoarded this material? It could have been one very efficient thief, but the going theory which seems more likely is that this hoard was collected by the victorious army and, at the end of the battle, pooled and saved for when they planned to return. In this case (lucky for us!) they didn't come back, but if this was how armies usually operated, it suggests that already by the 8th century we were seeing the kind of centralized control of plunder that Beowulf idealizes, with the king or lord presiding over a collective pooling and redistribution of the wealth stripped from a defeated enemy.

Another important site is the cemetery at Repton. This cemetery contains hundreds of bodies that certainly belong to one of the invading Viking armies (careful studies of their teeth and radiocarbon age have been conducted, which show us where they came from and when they died). Repton has a very careful 'warrior' burial that may, we could guess, be a chieftain; but the rest of the bodies are buried without weapons or other wealth. These armaments must have been collected: but by whom? The victors, or their own comrades? Their leaders, or their peers? I don't think we can say.

What does this all add up to? I think we can say with some certainty that, by the time that Cornwell describes (the late 9th century) there was certainly an idea that loot should be doled out by the king/lord. The actual practice is much harder to scry, however, and probably did include a fair bit of looting. I imagine, however, that few peasant farmers would have been allowed to walk off with the better treasures -- I do not doubt that their masters would have insisted that fine swords and mail go to more appropriate homes. There's a famous story, from a few centuries earlier and on the Continent (in France), in which King Clovis of the Franks sees one of his men with a beautiful pot that he captured from a sacked town. Clovis says that he had already taken his king's share of the loot, but wonders if he could also have this vase; the man refused, and to prevent Clovis taking it by force, smashed the pot. Clovis says nothing, but the next year when they meet again, Clovis grabbed the man's axe and tossed it to the ground. Confused, the man bent to pick it up, and Clovis smashed his head just like the man had smashed the pot.

Ie, no matter the time and place, the rich and powerful always tend to get the loot they want, else there are consequences.