And if the British wanted to avoid the war at any cost, why exactly did they give unprecedented support to Poland in 1939 IOTL?
Because at Munich we had conceded all halfway reasonable German claims - and within less than six months they had kicked sand in our faces by tearing up the agreement almost before the ink was dry, and seizing a country to which they had no shadow of a claim on any basis of self-determaination.
As a result, when Hitler came back with claims against Poland, no one took them seriously. It was taken for granted that they were merely pretexts for further expansion, and that Danzig, if conceded, would be just a preliminary to taking the rest of Poland, as the Sudetenland had been in the case of Czechoslovakia.
As a teenager in the early 60s, my local library still contained quite a few books from c1938 supporting German (and Hungarian) claims against Czechoslovakia. OTOH, I don't recall any supporting them against Poland - though German claims to Danzig were on paper at least as good as to the Sudetenland. Attitudes had shifted.
In 1938 the issue was the justice of German claims. In 1939, after the fall of Prague, the issue was Hitler. Of course it always had been for some, but until March '39 they were a noisy minority. After it, far more people, and soon a majority, came round to their position.
The final straw was probably the Molotov-Ribbentrop Pact. Till then, some people on the right had seen Hitler as a lesser evil to Communism, and even a barrier against it. Now, the anti-Fascists and anti-Communists found themselves on the same side - probably to the embarrassment of many of them - and the country was pretty much united.