They did, sort of. Right after the war basically every major power looked closely at the Mkb-42/MP-43/MP-44/Stg-44 line of rifles. The most notable example was the British EM-2, a bullpup rifle firing the .280 British cartridge, which was actually slightly above the 8mm Kurz used in the Sturmgewehr. In the mean time, the Belgian FN factory at Liege had worked on the FN FAL, which was initially also chambered in .280 British. However, the American military, which had a tradition of individual long range marksmanship, claimed that the .280 British cartridge was too weak, and insisted instead on the 7.62x51mm cartridge, which was more akin to the .30-06 cartridge (slightly shorter, but same energy) the US used throughout both World Wars. Due to America's military and economic might at the end of the war, it was able to convince the rest of NATO to adopt the 7.62x51mm cartridge. The FAL, being a more traditional, robust design, was able to be made in the American calibre and versions were made of both calibres during trials, and was adopted by most 1st World-aligned nations, including the British, while the EM-2, which was less adaptable, was abandoned.

There were other smaller projects, for example the Spanish CETME which had their own high velocity cartridge, which was also adapted to 7.62mm NATO to conform with standards. The CETME eventually became the H&K G3.

In broad strokes, it could be said that the reason for the former Allies not adopting an assault rifle was institutional culture and stubbornness within the US military. The British were later proven right and the M14 become the shortest serving standard infantry rifle, being quickly replaced by the M16 in an even smaller 5.56mm cartridge.