I don’t know a lot about the area but here’s a try: in 1932 Oman got a new sultan, at the time he was young, only 22 years old.
Let’s flap some butterflies and suppose he gets radicalized in his youth, or just plain drinks some stupid juice. Shortly after taking over he pisses off the British real bad, maybe nationalizes the oil reserves and shuts them out, whatever.
Some republicans inclined militants reach out for British backing to overthrow the sultan and get it. The plan works fairly well and as of the next year Oman is again British aligned again but with a republican government now.
Oman stays a British “ally” (maybe even puppet state” and is comfortable open for business through the 30’s and 40’s.
I’m the 30’s Britain takes a more aggressive stance in border negotiations with Saudi Arabia, insisting on the 1913 and 1914 Blue and Purple lines as established with the Ottomans (see map). This leads to some skirmishes involving the Saudis, British colonial troops, and the Omanis but eventually is accepted. Within the British-Omani zone of Southern Arabia Oman is delegated the administration of the entire interior and most of the southern coast east of Aden.
When decolonization roles around the UK is fearful of small states falling to soviet influence (and public opinion views the local princes and sheiks as despots) so instead of granting independence to the gulf protectorates and Aden they are merged with Oman (already a reliable British ally for decades at this point), forming the Republic of South Arabia.