In March 1939, soon after the unilateral annexation of the remnants of Czechoslovakia, the UK and Poland signed a guarantee to the independence of Poland. When Germany invaded Poland in September the Allies, though, did nothing to stop Hitler; did they really want to stop Hitler or declared war mainly because of treaty obligations? To further specify my question: would have the Western Allies intervened if in March Poland had not signed the guarantee or if it was invaded instead of Czechoslovakia?
I can speak for France and Germany, and possible only the two, in answering this question, since I know more about them than I do about Britain.
In the case of France, the act of protecting Polish Sovereignty is a diplomatic reversal for her in the context of how France approached conflict with Germany. In WW1, when Germany attacked France through Belgium, it was Russia that came to France's most significant aid by invading Germany from the East. The Eastern Front of WW1 was possibly even more crucial to helping France in the war effort than Britain's continental contributions, and the threat or use of Russia as an effective counterbalance to Germany was favourable for the French. However, with all of the internal problems of Russia, their turn to Bolshevism, and the practice of intervention of the Allies, France could no longer depend on Russia for aid against German aggression. In Russia's absence, France was left with picking up smaller but more numerous allies in Eastern Europe, all of them descendants of lands lost by Germany, Austria-Hungary, or Russia herself.
Combined, the military strength of the Eastern nations such as Poland, Czechoslovakia and Yugoslavia was actually quite impressive; Poland has already proven her resilience and tenacity by successfully defeating the Red Army once, and Czechoslovakia's military industry was already world renowned as the Skoda Works was the second largest in the world and their guns and tanks were of high quality. Czech fortifications were first class and their army was well trained. The problem laid in that many of these smaller Eastern nations were more likely to feud with each other before combining in one huge front, and their smaller, weaker status reversed the situation for France; instead of Russia coming to France's aid in the case of German invasion, it seemed more likely that it would have to be France that'd have to come to the aid of the smaller and more exposed Eastern nations. The need for allies against Germany made the alliance-building diplomacy necessary for France, but with almost typical "Gallic arrogance", France found herself almost always asking "How can they help us" rather than "How can we help them?". Ironically enough, the creation of Poland, instead of being a buffer state that protected Russia from Germany, had become a state that prevented Russia from coming to the immediate aid of the West as it was in the way (this is all assuming the USSR was willing to aid the west, but the Allies distrust of Bolshevism alienated all of the possibilities of the Soviets working directly with the Allies despite optimistic showings of the Soviets willing to work with the West). The creation of the Maginot Line and the defensive mentality of the French Army effectively anchored France's military prowess to almost exclusively just France and Belgium by default. France's primary hope of resisting Germany on the offensive became marching into the Rhineland as they have done in 1923, to fight Germany there, but the 1936 reoccupation of the Rhineland shut France's primary and only hope of taking on the offensive. Eventually, this fatal contradiction between France's diplomatic commitments and posturing and her actual military thinking was one that Hitler was very keen at exploiting to great effect.
This is where France's diplomatic policy saw an almost acute split with that of it's military policy; the former was optimistic of action against Germany, whereas the latter since the end of the Great War and all the way up to 1939 was only growing more and more defensive in nature, putting into question France's willingness let alone capability of taking on the offensive in the case of war.
Germany's re-occupation of the Rhineland in 1936 was perhaps most telling of France's diplomatic paralysis self-inflicted upon herself by her own thinking and by her reliance upon Britain. By all accounts, on paper and in reality, France was the most powerful nation in Europe as of 1936. Germany had a mere 3 Batallions to cross the Rhine and primarily unarmed planes flying over to posture a great army when they gambled the reoccupation. A French resistance would have seen Germany swiftly defeated. But, Britain, preoccupied with Italy and her adventures in Abyssinia, saw the issue as France's problem and opted out to not act. For France, it was in her strategic interest to stop the Germans, but Britains inaction seemingly gave the political and military elite their own excuse for inaction. France's Commander-in-Chief, Gamelin, pointing at Germany having a million men under arms and 300,000 of them already in the Rhineland (a massive over exaggeration by all accounts), passed on the responsibility of action to the Politicians, who were preparing for elections in six months. Gamelin toted that the French Army was under-strength due to the reduction in military service, and that action would have entailed general mobilisation, the latter of which in particular spooked politicians. With only six months until new elections, the politicians found themselves unwilling to sanction action against Germany without Britain's explicit aid, and with their slew of excuses, blamed Britain for inaction and sat on their hands as Hitler pulled off his most daring gamble ever. The most drastic consequence of the 1936 was Belgium opting out of the Franco-Belgian alliance and returning to strict neutrality, effectively further neutering France's ability to swiftly act against Germany.
By 1938, it seemed that France and her Allies seemed unwilling to defend Czechoslovakia or her allies if it meant all-out war with Germany. They were (or, at the very least, France) too war-weary and immobilised by their own defensive doctrines. The French urged the British to find a peaceful solution to the Sudetenland Crisis in 1938, which Chamberlain infamously did so with appeasement. Prior to this, it's likely that if Germany made the same demands against Poland first, a similar act of appeasement would have occurred. However, Germany's annexation of the rest of the Czech Republic by 1939 sobered the Allies in time for Sept 1939, where the line had finally been drawn in the sand, definitively, far too late.
Main Sources: