According to the book Nasser's Blessed Movement by Joel Gordon, the US army once considered shifting regional command of American troops to either Cyprus or Gaza when starting to participate in Joint Middle Eastern Command in Cold War exercises in 1951.
Suppose they go through with it and a new base is formed in Gaza, then under Egyptian control. How would this affect regional policy and US military/political concerns? How does Israel react? Or Nasser, assuming the Free Officer moment continues with its historical trajectory?
Answering the OP’s question in turn- farouk grants this, Britain feels a little intruded upon. It’s a tough sell for the US to be able to stay in position after Free Officers take over. The only appeal of the Americans is they are not British and might hasten their departure.
Gaza is the only terrritory under Egyptian administration I could see them accepting a western base. They won’t be giving the US based on the canal.
MB, at least Qutb’s faction, will not be happy at all.
How will Israel react? Carefully.
Would fedayeen try to jump out from under the Americans to raid Israel?
Egypt might set up a rule like Saudi Arabia pre-62, no American Jewish personnel.
American service people would hear a fresh and raw refugee perspective, but the US troop presence would provide job and business opportunities, and also “morally corrupt” the place.
By far Christians are more numerous in the West Bank than Gaza, but I think there would be some.
I don’t know how strictly Egypt controlled population movements in and out of Gaza and how strictly they enforced any controls. If the controls or enforcement are loose I could see some Palestinians in UNRWA camps in Lebanon, Syria and Jordan moving to Gaza for jobs.