Although Churchill was not the Prime Minister until 1940, his voice did have weight behind it in the House of Commons and the House of Lords. Even as Chancellor of the Exchequer, Churchill was an important figure in British politics. I'm not a professional when it comes to politics, but even in Europe, the political atmosphere was volatile. The Treaty of Versailles basically ensured that there would be a second World War. Like Kermit said, Both France and Britain knew that they couldn't save Poland. But at least they could have offered a more amicable outcome instead of turning their backs on Poland.
To blame one person for the fate of a country is wrong. I would never say or agree with anyone that Churchill was the reason why Poland did not receive the aid they were promised. I feel that it was a combination of many things that caused the ultimate fate of Poland in 1939. That being said, the Pole did prove that they could defend their country till the bitter end.
Below is the Oath that all Polish military personnel took in 1924. (There are many variations. This is one:)
I swear to God Almighty my faithful allegiance to my Fatherland, Republic of Poland. I swear always to stand by the military banners, to uphold the constitution and guard the honour of the Polish soldier, to be obedient to the law and to the President of Poland, to faithfully carry orders of my commanders and superiors, to keep the military secrets, to fight for my Fatherland to the last breath in my breast, to always behave so as to live and die as a true Polish soldier. So help me God! Amen.