Too risky. See e.g. Burdet, Encounter with the Middle East. Because of Egyptian military deficiencies, which took time to rectify, 1970 was thought to be the earliest confrontation date.
There was also Jordan, Syria, and Lebanon. Though if other countries join the UAR would be more formidable.
The wisest course would've been to strengthen the UAR to the greatest possible degree before contemplating war with Israel.
Definitely better for Nasser to get other countries into the UAR before a war with Israel.
Provided regime change in the gulf was the work of internal forces and subsequent unity voluntary, NATO wouldn't have been ina position to do much since these are soveriegn countries. Maybe the CIA would try something.
Would NATO have considered creating internal strife?