It's difficult to make an absolute negative statement. That being said, I've never heard of any units in the allied armies that were specifically dedicated to recovering lost/discarded weapons or munitions after a battle.

There were a lot of camp followers who would loot a battlefield looking for this gear in every theater of the war. The African theater in particular was full of such looters. Bill Mauldin mentioned that the North Africans he encountered had particularly 'sticky fingers.' And do you remember one of the first scenes in Patton? In the aftermath of the Battle for the Kasserine Pass, the area was crawling with natives looting the bodies looking for gear.

In the US army today it's a common SOP to assign a particular fire team/squad (depends on the operational element size) to sweep a freshly taken objective for enemy weapons or whole caches.

Following the battle of Normandy the 59th (Staffordshire) Division was broken up so as to provide replacements for casualties to other units in 21st Army Group (Anglo-Canadian, Polish and Continental formations). However, it was not completely cannibalised, its artillery was retained as 59th A.G.R.A. while 197th Brigade was retained as a 'battlefield clean up' organisation. It was responsible for recovering corpses and body parts, the clearance of the debris left in the Falaise pocket and other battlefields in Normandy and the Northwest Europe campaign as well as sorting through equipment- the serviceable and repairable from the dangerous and discardable.

One source that confirms that it happened at least occasionally: In one of the war correspondent Ernie Pyle's columns he describes (in Normandy) a detail of several men tasked with rebuilding scavenged rifles. They would disassemble them, discard the broken parts, clean the usable parts, and put the parts in piles. By the afternoon they would start assembling new rifles from the piles of parts. So there must have been some more-or-less formal system for bringing the rifles back to this detachment.

(If that sounds like a boring column, it's not--like the best of Pyle's work, it's quiet and haunting, like when he describes a wooden rifle stock whose previous owner had inserted a picture of his sweetheart into a hollowed-out space and carefully placed a watch glass over it. The stock went into the pile with all the others, got reassembled into a new rifle, and soon afterward, no doubt, a soldier was issued a new rifle with a picture of another soldier's sweetheart--a soldier who was very likely dead.)

Armored units of all combatants had squads dedicated to retrieving and repairing broken-down tanks, tank destroyers, self-propelled guns etc. and making them fit for combat again. They usually employed purpose-built armored towing vehicles to fetch the disabled vehicles (often under enemy fire) and transport them to repair workshops.

The task of these guys was extremely dangerous and often pretty horrific, since they frequently had to clean out blood and charred corpses from the damage vehicles.

In the book "With the Old Breed" by E.B. Sledge (this book was also used as source material for the Pacific, and the author is one of it's subjects), Sledge speaks of how after the Battle of Okinawa, his unit and others were ordered to scour the battlefield. They were ordered to pick up brass, rusty weapons from both sides (breaking the stocks of the Japanese Arisaka rifles) presumably for scrap, as well as looking for bodies.

He also touches on this in his other book: "China Marine"

What about other wars? Like the Civil War?