It would have expanded to include Iraq and Libya most probably. Suddenly Mideast politics change substantially.
Three things for sure one Israel be war this country until to day no reason make painful comprise for peace with Israel. Arab Israeli severing in Israel army this point well.
The USA and USSR pull deep into Middle East it self. Since this country like turn USSR while good number other Arab states turn USA. Possible even World war III.
Last but not least Iraq and Iran war never happen Iran like conquest by this country.
Iraq's leadership was generally composed of pragmatist nationalists like Arif and later Saddam, I really don't think they've have gone for rule from Cairo, Hell, there's a very strong possibility that the creation of a powerful United Arab Republic would lead to the creation of power blocs against it.
The OP basically makes the UAR more equal. Even without the OP, you always ignore al-Bakr...
The real power behind that particular throne was Saddam, who likely would not have cottoned on to that particular scheme.
Towards the end, yeah. He only didn't go for it because it took him down a couple of notches.
Yemen might join. They considered it OTL, but decided not to.