The best chance for the British and French is that they wait until after the US elections (Nov 6). It would only be something like a week delay. The US may still publicly be against the invasion, but Ike may not take actual measures to compel the British to back down. Since the invasion would occur after the Soviet invasion of Hungary (Nov 4), it might be seen as a kind of quid pro quo retaliation against a Soviet proxy and therefore be more accepted by some of the international public.
I think the British and French would be more successful in regaining control of the canal, perhaps under some form of international supervision with Egyptian participation but not control, but not successful in removing Nasser. Nasser would receive a huge blow to prestige and Nasserite version of Arab Nationalism be much less attractive, but I don't think the monarchy is coming back. At most, one of the other 1952 coup leaders will step up and replace Nasser eventually, although Nasser will likely still have a role unless he is totally removed from power in palace intrigue. Any long term benefits to the West really depends on what happens in Cairo behind close doors as a result.
Everything depends on whether the Anglo-French forces completely control the canal zone and defeat Nasser's "people's war". If so, Nasser has to come to some sort of political deal - perhaps with US mediation. If not, then it'll turn out in defeat regardless. But if Israel controls all of the Sinai, and Britain & France controls the canal zone (and shipping can go through), then they can hold out indefinitely as long as they retain popular support at home and can fend off international diplomacy against them. Both are a big if, but is possible. If so, then eventually Egypt has to make some kind of deal.
There will be no peace deal between Israel and Egypt although the Sinai might be completely demilitarized (and if there is international control of the canal, there may never be Egyptian troops allowed back there) and Gaza occupied. Israel won't get to keep the Sinai although strategically they may never have to worry about an attack there ever again.
Arab Nationalism will likely take a more Soviet turn, and the Ba'athist version gain dominance over the Nasserite version. ITTL, the attack is probably blamed for an eventual increase in Soviet influence in Syria and Iraq, although compared to OTL it is probably almost no different.
British and French prestige in the region would be increased although at the price of greater hostility. But this will not confer any real advantage to either of them.
The US is put in a very hard position if diplomatically it supports Egypt, but takes no practical action to force the British out. Things could go right that the US looks good in the Arab world, but a lot could go wrong. Again, much depends on what happens in Cairo during the crisis.