It's a reasonable request. Article 50 is triggered when a country informs the EU it plans on leaving. After that they have two years to negotiate the arrangements of the withdrawal and a framework for future relations. There is no reason why Great Britain can't invoke Article 50 immediately and start the clock running.
As for the prime minister part, France has no say in that matter. The Conservative party has to select a new leader first, and that should be done well before October but certainly not within days.