The 22K is simply a current limiting resistor and the 10K is a pull-down load for whatever is providng the serial signal so it should work; the AXE027 uses 0V/5V signalling and works across the entire PICAXE range with the resistors in place.
As always, the best course is to prototype the configuration, resolve any issues there may be before committing to a board design, PCB or strip-board, as things will be much easier to fix when on breadboard etc.
You can test by adding a 3.5mm jack plug to the sending PICAXE and plug that into the download socket of the other.