Obviously, there are "grave robbers," people who steal possessions (and/or the bodies themselves) from people's graves, but that's not quite the same thing.
A catch-all legal term for such theft is "estate looting," one who performs such looting being an "estate looter." This would apply because if the stolen items had been left on the bodies, they'd belong to that person's estate, go to that person's heirs or next of kin. But that's a catch-all term, not one specifically related to the battlefield.
"body looter"
In the vein of "looter," another term I've heard in the context of the Battle of Gallipoli is "body looter." While I couldn't find those examples, I did find others where it's used in battle contexts:
"Tiredness gnawed at him; he cared little for the body looter, but as
he turned back towards the streets a window shattered amidst a woman's
screams..." -Master of War by David Gilman
"He looks for his watch and then he remembers the body looter who was
taking it off his wrist when he was lying in the field." -The Window
at St.Catherine's by John Dobbertin Jr.
"I’d been in the Golden Horn before: with Cal, on jaunts to recover a
young fallen before the body-looters got at him. As O’Connor and I
walked past the dilapidated buildings, I felt Cal’s absence all the
more keenly—and it wasn’t a body-looter..." -"The Inaccessability of
Heaven" by Aliette de Bodard
"He saw his enemy standing over him with his wrist-watch in his hand... "So you are a
body looter as well," he said; "you rob the fallen." -The Lion and
the Adder: A Story of the South African Rebellion by Leigh Thompson
(In the second example and the fourth example above, each does refer to taking
from the dead, the given subject "he" being a different male than the male
that "his" refers to, each account being a male soldier speaking of
another male soldier who is deaceased and a belonging of that other male soldier who is deceased,
which, by coincidence, in both cases is a watch. That said, I'm not sure there's much distinction betweeen stealing from a soldier who isn't yet dead or who is too gravely wounded to be conscious or to prevent the the theft but ends up surviving, so I would imagine the term may be able to apply in those cases, too, though I don't have any examples to support that. On the other hand, maybe not as "body" in such contexts is rather suggestive of being deceased.)
While a Google search shows "body looter" is used mostly in war and battlefield situations, there is a spattering of examples where it's used in other contexts, but given how the preponderance of its usage appears in battlefield situations, it very well could be that it's battlefield term or a term born on the battlefield but has been appropriated for use in those other contexts. I can't say for sure because I can't find any etymology or really any research at all on the term "body looter," just a plethora of examples of its usage.