Mine has a complex GKN gearbox between the engine and the generator (and front wheels). (There's a lovely animation of it working here:
https://www.youtube.com/watch?v=rntjceP-XPE )
The engine is started by having power from the traction battery, supplied to the generator, which turns the engine over.
The result is instantaneous, if you touch one of the buttons ('Charge' or 'Save') that manually starts the engine, the engine goes 'vroom' immediately. There's no sound or sensation of cranking or delay. (It happens so fast, you don't even get to remove your finger from the button before the engine starts.)
Similarly, if I'm driving in traffic and accelerate beyond what the battery and drive motors can provide on their own, the engine instantly starts, with no cranking etc.