Middle East leaders quite frequently survive military defeats. Nasser himself survived 1967 even though the Suez Canal wasn't reopened until 1975, long after his death. Gadaffi survived the loss of the war with Chad. https://en.wikipedia.org/wiki/Toyota_War Saddam Hussein survived the failure of his 1979 invasion of Iran and his loss of Kuwait in 1991. It took the Americans actually physically occupying Baghdad to end his rule. The British and French weren't willing to do that with Nasser in 1956. Their objective was to make him fall--yet they weren't wiling to engage in an all-out war to do so--not even all-out bombing, let alone a seizure of Cairo. This attitude betrays an underlying weakness in the British and French which the Egyptians understood--just by keeping Nasser in power, they could win a "victory." And again, you have to ask, who is going to overthrow him? He had purged the military quite effectively. And the old politicians were discredited.Nasser losing the Suez Canal, losing Port Said/Port Suez (and likely Ismailia) to Franco-British forces, and being able to threaten Cairo, a lot of his support is going to evaporate.
And remember that the operation had been controversial in the UK from the beginning. It wouldn't become more popular with petrol rationing reinstated (sabotage of the pipelines), with more British casualties, with publicity about Egyptian civilian casualties, etc.