There are any number of ways this could go.
First, Chamberlain was firmly convinced of Hitler’s good will; it took the seizure of Bohemia and Moravia after his guarantee of their integrity at Munich to convince him otherwise. So France would likely have been left without British support if they took a hard line. (I’m not prepared to speak of the probability of their actually doing so.)
However, the Ardennes modification of the Schlieffen Plan that led to the collapse of France in 1940 had not yet been proposed to Hitler in 1938. So presuming war broke out, you would have, initially, a World War I static front in place in the West – until someone, presumably a German general, figured out how to make a breakthrough with tanks and/or the Luftwaffe. (There’s a lot of detail in the histories about the integrated assault techniques of Germany vs. the French view which expected tanks and air to play extremely minor roles in combat, and had so structured their own armor and air as to ensure that they would not have significant roles.)
Against this, however, there’s the argument that France and particularly Britain were nowhere near ready to fight in terms of materiel and troop effectiveness in 1938, and that, despite Germany ramping up at a higher rate, the year’s respite is what saved them, because it meant they had time to get something ready to fight with. I don’t have enough knowledge of circumstances and conditions to be able to talk effectively to this point.
Do not forget, too, that while Czechoslovakia was a relatively small country, it was armed to the teeth, had strong natural defenses, and an army of some 25 divisions to defend it with. So invasion of Czechoslovokia would not be a cakewalk, even for Nazi Germany – especially with France on the other side making it a two-front war. And the Soviet Union was prepared to back Czechoslovakia to the extent they were willing to accept help from it – for reasons obvious in hindsight, Benes and Mazaryk did not trust the USSR in the slightest.
But finally, the German General Staff was so opposed to going to war at the time of Munich, believing Germany was not yet ready, that they had a plan in place to overthrow Hitler & Co. – but Chamberlain’s negotiation of a peace prevented the necessity of doing so, and instead “proved” to them that Hitler was a genius, in that he’d taken a calculated risk and succeeded at it.
So you have three scenarios, and several supplemental variables to be dealt with:
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France and Czechoslovakia go to war alone, with initially indecisive results. How this plays out depends on a multitude of factors – can 25 Czech divisions fighting in fixed defenses hold off what Germany can afford to throw against them while combatting nearly as many French divisions as its entire army? How long will it take the Generalstab to realize the French weakness in use of armor and air, and capitalize on it? Will the French ever decide to gt a general who will go on the offensive, and what will the results be?
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Will the comparative degree of preparedness be sufficient to cause Germany to overrun France quickly, like before the end of 1938?
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Will the failure of negotiations and ensuing declarations of war provoke the Generalstab to launch their coup? Will it succeed? Will Germany erupt in civil war, Nazi loyalists vs. old-tradition Army loyalists? What will France and the Czechs do if this happens?
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Given any of the above scenarios, what will Chamberlain’s Britain and Roosevelt’s America do in response? Remember that Churchill is still a pariah to most Brits at this point; the events that proved him right have not yet happened. And America is still largely isolationist, and the events that led many Americans to become interventionists have not yet happened.
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What will Stalin do? Support the Czechs to the extent they’ll take his help, certainly – but they’re not about to let his troops cross the country (and in any case, Poland and Romania block his access to Czechoslovakia). But in this upheaval, he may see his way clear to a power grab of the sort he actually pulled in 1940, with the Baltic States and Bessarabia (i.e., today’s Moldova).
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The Far East is not a major concern yet. Japan is still immured in China, and the events leading to a war leadership have not yet taken place. But what impact this will have on the E.T.O. is an interesting question.