“Can you see anything wrong with this?”
I think so.
The push-pull circuit in your diagram is supposed to work with a balanced PSU. That is, a PSU that has a ground, a +ve rail and a –ve rail. The output from the emitters is supposed to go through a load to ground, half way between +ve and –ve. The input to the bases is supposed to vary above AND below this ground potential. The load supplies the base current in this sort of emitter follower.
Your load in this circuit is a mofet gate, so almost no current is flowing through the base-emitter of the upper transistor when the input is high. And when the input is low, the base voltage never gets below the output voltage so the bottom transistor can never turn on.
Also, emitter followers always have the drawback of the output never being able to get within around 0.7v of the rails.
Unless you are switching very quickly, there should be no need for a gate driver anyway. The PICAXE can supply about 20mA sink OR source, so the gate will be driven both ways anyhow. If your PWM frequency is too high, and the MOSFET is overheating, just reduce the PWM frequency. I forget the reference just now, but search the forum for ‘prescaler’ or some such. It’s just a simple POKE command.
EDIT:
Oops. My bad. The PWM output can only source current (output high), so you will need to fit a resistor between gate and source to pull the gate back down to zero. I suggest something around the 1k mark.