I have not done any PICAXE work with 4 to 20 mA signals but from the hardware perspective if you think about how PLC analogue input cards typically function, they in fact have (or did have) a 250 Ohm resistor across the input.
With the 4 to 20 mA current flowing, this gave a voltage of 1 Vdc at 4 mA and 5 Vdc at 20 mA.
Then you can just use the standard ReadADC command.
Which weather station equipment are you using?
Much of the equipment I have seen details for in the past is based on 1-Wire comms.