The first flag of which nominally independent state (1921 - 1944) featured a Kholro, a Buddhist prayer wheel?

The Tuvan People's Republic. Now then, there are a couple of caveats to this answer, the Uryankhay Krai which was a Russian protectorate that immediately preceded Tannu Tuvan People's Republic (1918 - 1921) had a blue flag with a Kholro on it. The first two flags of the Tannu Tuvan People's Republic that also featured a Kholro (1921 - 1926) however both of these were considered "unofficial". The first "official" flag (1926 - 1930) and all subsequent flags do not feature a Kholro.

On August 12, 1921, the Tannu Tuvan People's Republic came into being and the TPRP took control, immediately recognizing Soviet Russia as its overlord, and only the Soviets and Mongolia recognized its separate existence.

DISGRACEFULLY, the world looked on as Tuva ceased to exist as an independent sovereign state in 1944. The growing dependence of Tuva on the Soviet Union had been displayed after Nazi Germany invaded Russia in Operation Barbarossa in June 1941.

https://www.thenational.scot/news/19513472.republic-tannu-tuva-country-voted-independence/

History of Tuvan flags.

https://en.wikipedia.org/wiki/Flag_of_Tuva