The Polish-British Military Agreement in August 1940 led to the formation of the Polish Air Force under RAF operational control. However the The Polish Air Force (PAF) was under the direct control of the Polish government-in-exile and was the only "exiled" government to have such a military agreement with Great Britain.
It is certainly true that Poland was equipped, trained, housed and fed in Great Britain but it is also true that the Polish government in exile had to pay for the equipment and uniforms etc that they were supplied with by Britain out of the Polish State gold reserves smuggled out of Poland ahead of the German occupation. So it is not as if this war materiel and aid was gifted to Poland and if they were merely part of the British forces then they would not have had to pay for their war own materiel. However it would also be fair to say that Britain was at some point in the same boat as Poland during WWII in as much as it was a recipient of vast quantities of materiel from the US which it also had to pay for form its gold reserves in order to keep it in the war.
Back in the occupied Polish homeland there was a fully functional "underground state" that was under the direct political control of the Polish government in exile based in London. A good account of this is: "The Polish Underground State: A Guide to the Underground, 1939-1945" by Stefan Korbonski who was the last person to hold the post of the Government Delegate at Home.
There are many misconceptions about Poland and its history during WWII...even amongst us "Poles".
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