That early, Chamberlain can't be forced out. He only resigned when the Norway Debate revealed major weakness of support (and even then, he technically won it - but there were many defections from the Tories so it was obvious he didn't have the support he needed in wartime and it was best for him to go). No such thing this early on.
If he resigns for any reason (possibly health) that early, it is very unlikely that Churchill would get the nod. The next in line might be Halifax, but it might go to someone who wasn't associated with appeasement. Depending on the timing or the resignation, and if Chamberlain knew beforehand he'd resign, they might be able to arrange things so that Anthony Eden might be eligible for the premiership. If not, perhaps one of the others in the war cabinet. Churchill was seen as too erratic and adventurous to be trusted with being PM this early. IOTL he had several successes with the Royal Navy and a string of morale boosting speeches to advance him to the front, but September 1939 might be too early.
In any case, there is no change in the war for the first year. There will be no early front against Germany (Britain and France aren't ready to do so) to save Poland. They'll wait until the Germans attack in May 1940, and the Germans still steam roller them through.
Afterwards, who knows? Churchill will undoubtedly play a major role in any war cabinet, and perhaps might even become PM in case of a crisis.