The preceding events couldn't really have changed, so everything preceding the nationalisation would tick along as per OTL. Once we reach Nasser's action, however, we'll see a far more muted response from Gaitskell (in spite of his fears of Nasser being a second Mussolini, much like Eden did) and no rush to military intervention. The Americans still wouldn't be likely to throw their lot in with Britain and France, so there'd be a huge blow to British prestige in the region from the off.
This scenario, presumably, antagonises both the Labour left and the Conservative Party. Harold Wilson, Barbara Castle, Richard Crossman, Anthony Greenwood, Ian Mikardo, etc. were all proud supporters of Israel and considered themselves Zionists, leading to an even wider chasm developing between them and Gaitskell ITTL. The Conservative Party, on the other hand, will see itself more united than ever (with only minor dissent with the likes of Edward Heath, who tried to be pragmatic when it came to the Middle East) over Labour's "mismanagement" of imperial policy.
It's a good stick to beat Labour with so long as Gaitskell stays in power and can't shake the lingering disappointment of this alt-Suez, leading one to conclude that the Conservatives have a fair chance of beating Labour at the next election. Then again, the economy will likely be in even better shape come 1959/1960 with no run on the pound.