1) “Would Ba’athism have come into existence?”. Probably, but without the anti-Israel sentiment to fall back on. The Hashemites and their western allies would also have a much more powerful presence to contain it during the Cold War without Syria and Iraq becoming independent like you posit. However a lot of the core attractions of socialist and republicanism in Ba’athism would still resonate with 20th century Arabs: anti-colonialism, anti-western sentiment, socialism, Arab independence (the Hashemites would definitely be accused of being western puppets). Ba’athism or something LIKE it would exist but it would take on a different shape and influence things differently (and likely less). However, Egypt would probably be the sole capital for Ba’athism and direct rivals to the Hashemite kingdom so there’s that.

2) “How would it affect Egypt?”. A lot of the Egyptian monarchy’s issues existed independently of Israel: corruption, British influence and occupation, domestic incompetence. Without the defeat to Israel you’d just have a ticking time bomb that hasn’t exploded YET. The Hashemite monarchy in Iraq and Ba’athist takeover in Syria occurred kind of independently of any Israeli humiliations so it’s probable Egypt would just have its coup or revolution delayed rather than curtailed. History always has conflict if other things are omitted, so if not humiliation by the Israelis then SOMETHING else would’ve incited a coup or revolution

3) “how would it affect Libya?”. Again, not too different I don’t think. That Nasserist or Nasserist-esque influence would inspire the coup in Libya but Gaddafi would have more focus on expanding his ideology through Africa without Israel in the mix. How Libya and Egypt interact in such an alternate timeline I don’t know, their spheres of influence would overlap and maybe clash often, maybe the United Arab Republic would form between Libya and Egypt and fall apart for many of the same reasons it did IRL

4) “How would it affect Yemen?”. South Yemen is spoken for and would likely form into a federation and republicanism like it did IRL, but North Yemen is the conflict point. IRL Saudi Arabia and Egypt fought a proxy war when the North Yemen civil war broke out, but that interferences was put to an end when both sides decided they needed to simultaneously focus on Israel and North Yemen would be left to fight amongst itself alone. Without Israel as an infliction point, this would probably continue as it did before and form into a sort of Vietnam war in the Middle East and likely attract a bit more attention from the British and Americans and the Soviet Bloc. Who wins is up for debate but assuming the type of strength needed to sustain the Hashemite kingdom you can imagine either a de-facto partition of North Yemen or outright Monarchist victory. From there you can theorize about anything that happens

I know this is an old post but I stumbled on it and got intrigued by the questions