RC Timer Circuit (fundamental) Misunderstanding
The bit I don't understand is why V1 and V2 in the diagram are not 9v
  instantly (ignoring the voltage drop across R1)


Well if you ignore the volt drop across R1 then of course V1 and V2 (the same node) will be 9 volts the instant SW1 closes and infinite current will flow into capacitor C1.

So, ignoring stuff usually gets you the wrong answer.

The better answer is that after some length of time voltage V1 will be a steady DC voltage that is somewhat less than 9 volts. That is due to R1 dropping a voltage due to the potential divider formed by it (R1) and the resistance of the buzzer (RB). Let's say that (for the sake of argument) the voltage eventually reached is (say) 7 volts.

Then the actual voltage over time (v(t)) after the switch closes is: -

$$V(t) = 7(1 - e^{\frac{t}{\tau}})$$

Where \$\tau\$ is the parallel resistance of R1 and RB multiplied by C.