Hitler's plan was for war with France and Britain in 1942. He only saw war with Poland as a short war to blood his armies and obtain fresh plunder.
If there is no British guarantee of Poland, then there is no M-R Pact as Hitler only did that to scare Britain into not honoring its security guarantee. Without it, Hitler likely thinks there is no need for a pact as there will be no greater war (of course, he also thought the M-R Pact would prevent war with the West in 1939, and that didn't work out).
There are a lot of unknowns here, and much depends on what exactly happens as they could lead to different conclusions.
Would Halifax become PM? He was from the House of Lords. A more likely successor would come from the Commons, although it would have to be someone in support of the Munich Agreement at that time. That eliminates a lot of candidates - no Churchill, no Eden, no Duff Cooper. The other senior cabinet members were Chancellor of Exchequer John Simon (who had an unlikable reputation and was formerly a Liberal), War Minister Hore-Belisha (disliked as a Jew and also a former Liberal), Home Secretary Samuel Hoare (now infamous for the Hoare-Laval Pact), and Secretary State of Air Kingsley Wood (who did a good job, but was not yet prominent enough to be PM in my opinion). Samuel Hoare was a key member of Chamberlain's ministerial circle. He doesn't have the disadvantages of Simon and Hore-Belisha, and he is much more experienced than Wood. That would be my pick to be the next PM. He could be trusted with the high position and would be expected to follow Chamberlain's "successful" Munich policy. In general, Samuel Hoare believed in delaying war and aggression while rearming. So we won't have a very different policy than Chamberlain. However, at least in Wikipedia it says he believed more strongly than Halifax that peace was guaranteed because of Munich. However, it won't be very long until Hitler violates the agreement, and he'd need to be more realistic about the need to prepare for war.
But what does he do when Hitler marches into Prague? I agree with others that the PM (Hoare or Halifax if we still go with that) would need to do something about Poland to restore British credibility and prepare for war. However, whatever is done in regards to Poland and the Soviet Union is very important. Specifics matter.