You need a circuit will more gain, such as those built with an op amp (used as a comparator) or a 555 timer. A single transistor does not have enough gain to give a digital snap action in your circuit.
The circuit you built just delays the initial part of the RC exponential ramp, but once the zener voltage is exceeded, then the later (slower) part of the exponential starts to turn on the transistor, slowly as you have observed.
Edit: Another way to provide a snap action is to use an SCR in place of the transistor. The power must be removed from such a circuit to reset the SCR after it is triggered. Use a sensitive-gate SCR to minimize the required gate trigger current.