During WW2, Germany prominently made use of a variety of captured equipment, such as the 35(t) and 38(t), repurposed French Lorraine 37L and H35 and even Char B1s. They also appropriated, to my understanding, especially the Czech industry to continue the production of domestic military hardware for the German forces.
For some reason, very few such accounts exist about the Allied militaries. There are some isolated incidents of individual vehicles or aircraft captured and fielded or flown under the flag of the Americans, British or Soviets, but no large-scale appropriation of hardware or industry.
Perhaps the question is less relevant to the Western Allies, because the Americans and British just have to fight their way through France for the most part, before reaching Germany, and local industry here is rightfully that of another, if formal, Allied country, that of France, so in that sense, the Allies are appropriating French industry by giving it back to its rightful owner, France, and the French presumably use it for some minor military engagements. But in the East, the question becomes interesting.
When the Red Army enters Eastern Europe with its native military industry, there is no appropriation of factories for military purposes, there is no equivalent to the Czech or French tanks anywhere in the Soviet military registers.
Is my impression correct that this difference exists? If so, why was the approach so different? Were the Allies simply not in need of additional production capacity as badly as the Germans were? Would the local produce simply not been up to making any meaningful battlefield contribution so late in the war? Or was the progress of closing in on Germany simply too fast for any of these adaptions to be made in time? The Soviets certainly did not object to looting infrastructure in principle, as they did so across eastern Europe after the war, and also carried home large quantities of weaponry and vehicles from Germany as well, which then resurfaced in the Middle East and possibly North Korea. But during the war, it seems to have been immensely uncommon.
There simply didn't exist the need. Captured Axis vehicles were used in the Western Desert campaign but very often broke down without doing much good.
By the time of the Normandy invasion, the Western Allies were flush with equipment and vehicles that had been stockpiled in England for the invasion and more and more were arriving from the US since the threat from U-boats had largely been neutralised.
As you say, captured equipment did make its way into Allied hands but this was relatively small scale. The reason for this is that there existed virtually no pool of spare parts or expertise to maintain Axis equipment. Meanwhile, a knocked out Sherman tank could be patched and put back on the line within days and allocated to a replacement crew. There is no doubt that the Allies at times faced issues of supply but these never prompted the widespread use of captured equipment for use as a stop gap.
I can't speak as much about the Russians post 1943 but I understand their supply issue to be better and that they generally had enough to go around.
The Germans never had enough of anything because their war industry was notoriously poorly managed. The use of captured material only added to their woes as spare parts for multiple different vehicles were not interchangeable. Equipment and vehicles from Czech, Polish, Russian, French, Belgian, German, Dutch, British, Greek sources and more found there way into German service and all of these needed parts to maintain. Additionally, captured artillery and tanks needed different ammunition then standard German equipment and all needed to be produced, stored, transported and allocated to the correct place. For a country still heavily reliant on horse drawn transportation, it was a nightmare of logistics that the Allies never had to worry about.