Hi,
A problem is that I don't think there's a published schematic diagram for the AXE023, so something might not be "as expected". First, the board carries a L293D driver chip which is "unnecessary" for driving a servo motor and may be doing something "undesirable". In particular, that chip has a pair of "Enable" inputs which requires at least one to be held High to pass any input pulse on to the output terminal.
Finding the "fault" is really just a matter of checking whether all the circuit voltages are "correct", in particular the servo pulse and the presence of ALL the required voltage levels, at a sufficiently low source impedance. The method will depend on what "tools" are available, an oscilloscope is ideal, but a Multimeter is sufficient for most purposes, or even a LED with a series resistor and a couple of wires/probes may be sufficient.
First ensure that the supply voltage AND Earth connections are reaching all the required components, then follow the Servo pulse through from the PICaxe pin to the servo motor connector. The servo pulse is more difficult to detect, however a multimeter (on DC range) should measure about 8% of the full supply voltage (or 92% down from the supply), or a LED should shine less brightly, but still clearly. If the basic hardware seems to be operating correctly, then adjust the software, for example the Pulse Duty Cycle or the pin number, to check that all the drive signals change as expected. If you can identify something "wrong" or unexplained, then perhaps we can find a reason.
Cheers, Alan.