I sincerely doubt that Hussein is the only reason the Iraq-Iran War began... boundary disputes and oil reserves and religious differences (Shi'ite and Sunni) and cultural and linguistic differences will still exist and will almost certainly all add up to war. The Revolution had more to do with Western influences and the backlash against it.
We also seem to be forgetting that Jordan is a Hashemite kingdom (no matter how you transliterate it in English, it doesn't really matter!) We saw in the early 1900s with WWI that related monarchs (despite the view "the war will end when my cousins and I sit down") does not eliminate or discourage wars. You could see a King of Hejaz, if he defeats the Saud family, eventually try to take the Caliph title with the support of cousins on many of the other thrones, as the Hashemite Hejaz kings are descended from Mohammad himself.
For Iran/Iraq, hard to say. The underlining tensions already existed true, but there's a difference between tension and out and out war. After all both were technically allied by the Baghdad pact. It also depends on Iran. With a POD in 1920 there's no guarantee that an Iranian revolution would occur, nor that it would result in an aggressive Theocratic state. Hell the POD was before the Pahlavis even came to power, so we could end up seeing a Qajar Iran (near ASB I know but within the realm of possibility).
To Jordan and WWI, there's a difference between cousins and brothers going to war with each other. Not to mention they were all of the same family line. We never saw real tension between Jordan and Iraq during the latter's time as a Kingdom, so no reason to think that the same stable situation wouldn't exist TTL. As to the Caliphate, hard to say. OTL King Huessin's proclamation of his Caliphate met mixed reception and didn't stop his Kingdom's destruction by the Saudis. However, if the core middle east is entirely ruled de-jure by his sons, then such a proclamation could very much end differently. Of course its hard to guess how long the Hejaz would last if the Sultanate of Najd remains under the Saudis. I can't see the British backing a Hashemite campaign to drive the Saudis out as it would give to much power to one family without a counterbalance.
I think we're ignoring the Israeli elephant in the room.
How does the Arab-Israeli conflict go with Hashemite monarchs in Syria, Iraq, Jordan, and Arabia?
Really hard to say. Again with a POD in 1920 we might not even see a State of Israel be created in the first place or it could be quite different than OTL. However, from what I remember the Jordanians were the most moderate towards the Israeli state, so that could be repeated among the other Hashemite Monarchies. Or a more unified Arab League could see a quite different end to the 1948 war. But I don't know much about Israel-Arab relations during that era so someone more knowledgeable than I can probably speculate better.
A surviving Hashemite monarchy in Iraq also means that Iraq probably never leaves the Baghdad Pact. Iraq remains a conservative monarchy like Jordan and is relatively pro-Western. Since Iran is also a member of the Pact, Iran and Iraq are allies.
If there is also a Hashemite Kingdom of the Hedjaz, then lots of things change. You don't have the Wahhabists in charge of the holy cities (although likely they will still be in control of the oil rich regions in the east of Arabia). They would likely follow Hashemite policy of Jordan and Iraq (maybe even also joining the Baghdad Pact).
Assuming they stay in power (which is difficult, but not impossible - we have Jordan as a country which successfully thread the needle even as OTL Iraq is an example of failure to do so), this drastically alters the Middle East. The region is more pro-Western. However, it will have difficulty holding off the Communists, Ba'athists, and Nasserites. Much depends on what happens in Iraq, and if it devolves into a pro-Western dictatorship, or if some form of parliamentary rule survives.
In 1958, Jordan and Iraq formed a brief Arab Federation which joined the two Hashemite monarchies together. It was done as a response to the United Arab Republic under Nasser. The Arab Federation united foreign policy and defense of each country but left the vast majority of other domestic programs under national jurisdiction. It didn't survive the 1958 coup in Iraq and therefore only lasted half a year. ITTL if the monarchy survives, then the Arab Federation continues. It also likely includes the Kingdom of the Hedjaz.
It would be very interesting to see how/if the Arab Federation survives. With Baghdad, it controls one of the more important Arab capitals. It will also have guardianship of the holy cities. Iraqi oil also potentially offers some benefits for Jordan and Hedjaz, although Iraq will need to be careful not to alienate domestic opinion. Since the rulers of the three kingdoms share blood ties, it conceivably has a better chance of survival than the UAR.
In such a scenario, I think this puts a lot of constraint on Israel. It possibly puts Jordan into a more anti-Israeli position. It also makes Israel much less likely to antagonize Jordan which is part of the larger Arab Federation. The West Bank (and even East Jerusalem) might very well never belong to Israel. And even if there is a war scenario and Israel occupies it, the chances of a Hashemite-Israeli peace deal is much greater. The Palestinians will never have a chance for a state of their own, and they'll have to tolerate being part of a greater Hashemite kingdom. The PLO likely never achieves the status it does IOTL which also means their terrorist attacks never occur. I don't see anything like the murder at the Munich Olympics ever happening. And if the PLO never goes to Lebanon, then that has lots of potentially beneficial butterflies for Lebanon.
Thanks for the detailed response. First off, not to sure about the Saudis. From what I've read about Ibn Saud he seems like an all or nothing type of guy. Would it be possible for the Hejaz to crush the Saudi state and form their own version of Saudi Arabia? If not then I think we'd see multiple wars between the peninsula's two native powers.
To what Iraq would look like, hard to say. I'd guess it would depend on how long Nur al-Said remains in power. Under his premiership the Parliament was a rubber stamp assembly, so once he's out the Iraqi government could begin to transition into a more democratic or at least parliamentary state. Although either way the Monarchy would retain a large amount of power, though how its used would depend on Faisal II (we don't know much about him).
Second, considering the ambitions of Hussein ibn Ali which seemed to be shared by his sons, I could see a TTL version of the Arab Federation form earlier, like say the forties. Maybe in place of the Arab League. So assuming the Hashemites form an Arabian Kingdom, we'd have all the territory between Egypt and Iran under a single polity. Considering the oil wealth of these states OTL it would potentially form a middle power, rivaling the West and the East.
Third, at this point would Israel even be in the cards? I mean if the Arab League/Federation is more unified we could see the TTL 1948 war end the Jewish state before it properly begins. Although, at the very least Israel would remain in its UN borders while the rest of Palestine goes to either Jordan (if it exists here) or to Syria. Would Lebanon still end up as an independent nation here or would it be a state/province of Syria?
I may be mis-remembering, but didn't the Hashemites get the throne of Iraq after they lost the throne of Syria? Didn't the British throne them a bone after the French kicked them out of their Mandates?
Didn't all of this come about after they lost Hejaz anyway? If they kept that, wouldn't they have no reason to be in power in Syria and Iraq?
Yes and no. The initially Iraq was to Faisal's brother Abdullah in March 1920 but he declined (not sure why). Later after Faisal was deposed he was suggested as a candidate for the Iraqi throne and accepted.
As to why, Hejaz had the holy cities but not much else. Iraq and Syria were two of the wealthiest countries in the Middle east and sat on strategic territories. Why limit your family to one throne when it can have three or even four?
Different colonial policies, different regional alliances. Surviving Syrian Kingdom probably means no Hashemite Kingdom of Iraq, unless they can find another brother who never got a crown IOTL. Could change the course of WWII.
OTL there was a lot of musical thrones in 1920-1921. Originally Faisal was to be King of Greater Syria, which was modern Syria, Lebanon, Jordan and Israel. Abdullah was designated as King of Iraq at the 1920 Cairo conference but never formally ascended the throne. After the Syrian Kingdom was dissolved by the French Faisal went into exile in London and was later made King of Iraq in 1921 because of his apparent conciliatory attitude towards the Great Powers and based on advice from T. E. Lawrence, more commonly known as Lawrence of Arabia. In turn Abdullah was made the Emir of Transjordan, another of Britain's Mandate territories, later becoming today's Jordan.
So if Faisal keeps Syria then Abdullah becomes King of Iraq (the offer preceding the fall of Arab Syria). Then the only Hashemite Prince lacking a throne is Prince Zeid, OTL's later Pretender of Iraq. And even then he could end up as King of Palestine, at least potentially.
But not the "Hashimate" of the title.
Yes I know I misspelled the title. If there was a way to edit it I would.