Question

why the charge across the capacitor does not change instantansously

Answer 1

The voltage of one plate compared to the opposite plate jumps discontinuously. On the other hand, if you want to adopt a microscopic viewpoint and consider the internal construction of the capacitor, you could say that the voltage potential from one plate going through the air (or whatever the capacitor medium is) to the opposite plate does rise at a large, though continuous rate if the capacitor is charged.