Utilizing captured equipment was fairly common for all sides in WW2. Of course the lack of spare ammunition and spare parts often meant that usage was usually temporary. The Germans in particular utilized enormous quantities or captured equipment during the Second World War, foreign produced equipment was often given a designation suffix to prevent confusion with similar equipment and denote country of origin. Czech equipment was designated (t), British (e), French (f), Soviet (r) etc. The most famous examples of these would be the beute (or booty) panzers which made up a significant part of the German AFV stock, often serving in second line roles or doled out to their allies. Small arms were also captured and reissued in mass quantities, primarily French, Czech and Soviet equipment. Significant portions of the last ditch Volksturm troops were armed primarily with equipment of foreign origin. American small arms were generally availiable in much smaller number, as by the entry of the US helped mark the reversal of German fortunes. Provided a sufficient quantity of ammunition could be acquired, most German grenadiers woiuld be happy to exchange their Kar98k for a Garand or M1 carbine.

For the Western allies adoption of captured equipment was less regular as by the time the Allies started to capture significant stores of German war material they had achieved such an immense superiority of equipment that the added logistical headache of utilizing foreign equiptment generally outweighed the benefit. An exception to this would be the Western Desert where any equiptment, particularly armoured vehicles were highly prized and any vehicle regardless of origin would be eagerly used. Unofficially however, many soldiers would "acquire" additional weapons and equipment. German machine guns and submachine guns were reportedly highly valued by british troops as their own submachine guns were poorly regarded and the blistering rate of fire of German mg 34s and 42s provided a better "beaten area" than the slower, more precise Bren gun.

The Soviets also widely utilized captured equiptment, captured Panzerfausts were widely used and seceral captured Panzer IIIs were sent for a factory refit into Self Propelled Guns. Much like for the Western Allies issuing captured equiptment ad hoc during wartime was often not worth the effort. The massive stores of war material captured was reissued to Soviet allies post war leading to phenomena like Syrian Panzer IVs and Viet Cong mp 40s.

I hope this is of some use to you, when opportunity arose it would not be at all odd for Germans to use American weapons and vis versa, even if only temporarily