The latter was the case for the Czechoslovaks, with some minor caveats, and I imagine the Poles would operate under a very similar arrangement. According to a treaty between the Czechoslovak government in exile and the government of the United Kingdom signed on 25 October 1940, Czechoslovak armed forces would be "organized and deployed under British command". The status of the airforce was further detailed in an addendum from 16 December 1941.

However, by that time Czech airmen had already been serving in the Royal Air Force and the agreement basically only reflected the realities on the ground (in the air?). In fact, Czech squadrons had already been raised in the months before the signing of the initial treaty in 1940 – No. 310 and 311 in July, No. 312 in June, and only 313 came into existence after the treaty had been signed, in May 1941. You can see both the documents in Czech here and here, all translations mine.

Czech air units were "assigned to the RAF ... for practical reasons", but "acknowledged as units of the Czechoslovak airforce and a part of the Czechoslovak armed forces." Pilots and servicemen who could not be assigned to Czechoslovak units would serve in other Royal Air Force units. The agreement also stipulated that, following a transitory period under joint British-Czechoslovak command, the Czechoslovak units would be led by Czechoslovak officers.

Incidentally, plans for a Czechoslovak airforce had already been drawn up during the Battle of France earlier in 1940. But the swift German success there prevented its establishment, and only individual pilots – such as František Peřina – were assigned to French squadrons before the evacuation to England.

The addendum also outlined how Czechoslovak ranks would be converted and integrated into the British system, and how the Czech squadrons would reflect the British in other ways from dress uniforms to equipment. Just like you say, they were a part of the British order of battle, with whatever few caveats there were largely stemming from the political realities, such as the units reporting to exile government authorities as well as to the British chain of command, or joint British-Czech committees overseeing new recruits and the like.

Czechoslovak troops were financed by the government in exile using loans from the British government, which in turn were secured by the British being given use of what remained of the Czechoslovak gold reserve in the Bank of London for the allied war effort. Apparently, there was a difference in how the Czechoslovak and Polish forces were financed as the latter were not paid for by their government in this manner, but I do not know the details, just that there was supposedly a different system in place. Additionally, the issue of loan repayments and the gold reserve would remain unresolved until 1982 due to the political shifts in Europe after the war.