With regards Enigma, it is correct that the Poles were the first to crack Enigma enciphering, although the Enigma machine they cracked was not a military version but a diplomatic version with differed in detail from the military version. On the outbreak of war the Polish codebreakers handed their material to the French, who on the invasion of France handed it onto the British.
During the time the Polish codebreakers were at work the British at Bletchley Park had not been idle and had already made some of the same discoveries that the Polish team had, although both teams had found out information neither of the other had. The work done by both Polish and French codebreakers was to prove insufficent in cracking Enigma, the British made key breakthroughs not only in ways to speed up the cracking of the daily key, but for example, they also found an entiely new process of cracking the four rotor machines used by German u-boats and their HQ (The existing techniques proving incapable of this).
With regards the British not making good use of Polish codebreakers, this was an understandable although maybe misguided security precaution. It applied to anyone from an occupied country, including France and came about from a fear that pressure may be brought to bear on codebreakers from occupied countries by the fact that their families remained under German occupation.
The American involvement with regards Enigma was limited to providing the raw intecepts, Bombe production and there were some categories of messages that were deciphered in the US using British methods. There was no major US contribution on the methodology side of things of Enigma, or of other German codes although for Japanese codes the situation was entirely different (The US codebreakers tended to concentrate on Japanese codes).
Hope this helps.
