Well, if he dies when being overthrown from Syria then Abdullah likely becomes King of Iraq. Zeid may become king of Jordan but if he does I am not sure it would be the same way that Abdullah did. Abdullah basically moved in to Jordan with the idea of taking back Faisal's kingdom. The British did not want to have to spend money fighting for the area and by the time they realized it was a problem, Abdullah more or less controlled Jordan. They then recognized him as king but under their mandate and convinced him not to stir up trouble in French Syria. I am not sure if Zeid would do the same (he was only in his early 20's at the time). Though the British may find it expedient to name him King of Jordan later on their own initiative.
Hard to say what might change from that. I have heard the opinion that Abdullah was a better politician than Faisal, so, assuming he lives as long as OTL (or longer without a Palestinian nationalist having reason to shoot him) he may give the Iraqi monarchy a better basis for survival. Talal managed the transition to a slightly more constitutional system in Jordan, so maybe they can manage the same in Iraq. The politics of the nations are different however, and Talal may be different having never seen Abdullah assassinated, so it is somewhat up in the air. Zeid may do much the same in Jordan as Abdullah did.
If Abdullah is the Senior Hashemite and king of Iraq, that could change the interplay between him, the President of Syria, and Zeid as king of Jordan that were the source of some of the shenanigans around the Arab-Isreali war in 1948. I can not say exactly how however.
If Faisal lives longer he will probably be King of Iraq during WW2. He was starting to dislike Nur al-Said in his last years so it seems likely that al-Said's influence is reduced. On the other hand he is unlikely to support aligning Iraq with the Axis. He is probably too cautious for that. If he manages to get Iraq through WW2 without being occupied he is in a much better position to develop the country than IOTL. If Ghazi dies as OTL then he is also in a position to possibly influence Faisal II which is hopefully a good thing. Faisal being around during 1948 also changes the political dynamic though I am once again not qualified to say exactly how that would go.