“British troops did not save Poland; they did save Britain from the same fate.”
Britain (along with France) declared war on Germany to defend Poland; Germany had no resources to fight Britain (as borne out by the final result). Axis resources had to be dedicated to the East (war with the USSR was inevitable); subjugation/occupation of Britain was impossible.
After Munich, Britain had made the decision to declare war if Hitler made *any* more territorial grabs, in Poland or elsewhere. Letting Germany take Poland unanswered would have been tantamount to letting them take the rest of Europe, which would have had the effect of excluding Britain from any influence on the Continent and allowing the Axis powers to strangle Britain’s naval routes, placing Britain entirely at Germany’s mercy once they had finished with Russia and developed the atomic bomb. In 1939 there was no guarantee that Germany would attack Russia first. Hitler had been debating who to attack first for years, and did not definitively decide to attack Russia first until the end of 1940, after he had reached a tipping point in his concern that Soviet advances in the Baltic might preclude a future German attack on Russia and that Soviet control of Romania might cut off Germany’s oil supply. Hitler and his generals debated the issue of attacking Russia until Hitler finally convinced Goering (during a November 13, 1940 discussion following a Molotov visit to Berlin) on the grounds that Russian conquests would supply the food and oil needed to defeat Britain. As this indicates, attacking Britain was by this point a given for the German planners, it was just a matter of when. They had been subverting Britain from within since before the war in the hopes of installing a German-friendly regime. Hitler’s hope was to keep the United States out of the war long enough to take Britain before the U.S. could intervene. Without U.S. aid, and if Germany had held off on the Russian front longer than it did, and especially if Germany’s atomic bomb program had reached completion, subjugation of Britain was a real risk.
That said, I will grant that a strategic case could have been made to let Germany and Russia slug it out while Britain built its military strength elsewhere around the German Empire’s periphery, instead of committing to Stalin’s Second Front demands. Once the Allies had committed to a Second Front, Britain’s main goal was to stall the Allied invasion of Europe long enough to build up sufficient forces for victory, for fear that the U.S. was pushing too hard for an immediate invasion of Europe from Britain in 1942 before there was a sufficient build-up. Marshall was pushing for FDR to attack Europe from Britain immediately, arguing that if the Allies did not attack in 1942-1943 Germany would knock Russia out of the war. Britain feared an attack this early would have to be too small to achieve significant success, and ran the risk of letting Germany and Japan achieve enough naval dominance in the East to cut off Allied oil supplies from the Persian Gulf. So the British and Americans went back and forth on this, and meanwhile the Soviets of course pushed in favor of the Second Front and worked on influencing the Americans who leaned towards that option, especially Harry Hopkins and Marshall. The invasion that ultimately unfolded was the result of this tug-of-war over the direction of the Allied strategy.