Would Britain and Ireland be allowed to enter it earlier?
Their second application, in 1967, would likely not be blocked by France.
And who would be the next President?
That's a more tricky question than it seems. For one thing, at that point, the institutions of the Fifth Republic stated that the President had to be elected by an electoral college rather than by direct popular vote--in OTL, that was changed by de Gaulle with a referendum held in November 1962, but obviously if he gets killed in August, that decision will remain in limbo (the decision to hold the referendum would be taken in October, over the opposition of the National Assembly). Perhaps the Gaullists would still call for the constitutional reform, but it's anyone's guess whether they would prevail.
For another, de Gaulle's prime minister was the newcomer Georges Pompidou, who had just replaced Michel Debré. Pompidou would normally be the logical choice as the Gaullist candidate, but he may have been sidelined in favor of Debré who was the better-known of the two. The situation would have been as confused on the left, with men like Guy Mollet (SFIO: French Section of the Workers' International) and Pierre Mendès France (PSU: Unified Socialist Party) competing for the votes of the mainstream left, and the Communist Party probably going for a spoiler's strategy (its candidate would have been either Maurice Thorez, who had two years left to live, or Jacques Duclos; both were hard-line Stalinists who had not-so-privately criticized Khrushchev's denunciation of Stalin's crimes in 1956).
All in all, if one considers that the electoral college would have remained in place at least long enough to elect the new president, and factoring in a sympathy vote for the Gaullists, the next president would have been either Pompidou or Debré, and I'm leaning towards Debré for the aforementioned reasons. However, lacking the legitimacy and charisma of de Gaulle, his policies would have had to be more in the tradition of the Fourth Republic, and he couldn't have, as de Gaulle did, run roughshod over the National Assembly.