Glad the study is going well. It's a fun game (and very good training in intellectual rigour) to evaluate the biases of all teachers/speakers. We all have them of course as it comes with being human!

Question is, how soon post-war is Israel kicked out of the Commonwealth?  The  utter lack of oil from the Arab regions would demand something to be done.
I do not really understand the logic-basis of the question. The UK got its oil from the Caribbean, Mexico, Persia, Iraq via the TAPline to Haifa (which fed product mostly to Egypt, Greece, Turkey and Italy rather than the UK) and the USA in this era. The Gulf fields were not really on-line until the mid-late 50s IIRC and they mostly fed European countries and japan as it took off. UK refineries were a bit skewed to very heavy Venezuelan crude with lighter crudes from Mexico, US and Trinidad making up the balance. In postwar APOD, it's the UK which replaces Japan as the 'economic miracle country', because by doing so much better at the start of the war due to having to fight stronger oppoents, Japan does oh-so-much-worse at the end of it. The reason for that is that the absolute refusal of teh IJA to surrender will leave Allies with no choice (after pressure-mining shuts down their coastal shipping trade) but to smash the one thing they never touched in OTL - the Japanese rail net. They did not touch it because there's no way back after you do that - Japan's non-rural areas starve to death.

The brand-new refineries built to feed the reconstruction of Europe in the 50s were nearly all built to handle the light sweet Gulf and Persian crudes. It's not until the 70s that pan-Arab nationalism caused the oil shock.

But with a British 'economic miracle' from about 1953-55, the new British refineries will be built for Caribbean heavy crudes plus Mexican and Persian light sweet crudes. The Gulf ain't on-line yet, and why throw over existing suppliers when it's British investment which established their oil export industries in the first place? OK, that will change if they nationalise the British investments, but then the new Gulf suppliers have the opportunity to obtain a new market they desperately need as the world's in oil glut until the mid-70s anyway.

By then Israel would have been a fixture for a generation, with no history of the neighbours trying to invade them and triggering wars they then lose. We are looking at a Middle East where basically nothing much happens. Is there a basis for the Arab states to refuse to sell after the 70s? I don't know, but even if they did, the Persians would not agree (they hate Arabs), and Imperial Iran got on very well with Israel in OTL. And without the whole OTL pressure-cooker, Islamist extremism may not take off to the same extent, which means no Ayatollah Khomeini etc.

Which neatly avoids all the stuff we've discussed.

Cheers: Mark