Proper tools are best, but if you wish to attempt riskier homemade effort:
You might try cutting washers to a U shape to fill the gap between frame and gear, and pressing the axle out with a vice. You can use them to reassemble also.
Once the axle is flush to the frame however, you would need to use a small drift to drive it further. A hammer and drift used lightly may be possible at this point used carefully. So might the vice, if you can find something short and stable for pressing with. A drillpress might be used as a press, able to hold a drift straight . If both wheels are removed, putting the axle into a drillpress chuck, you may be able to pull the axle rather than press, but I'd start with pressing first, as more force is usually needed as a process like this begins, less as it moves along.
Also, if you have all new parts; wheels, bushings, gear, axles, and access to a modeling rotory tool (Dremel) and cut off discs (reinforced are best) why not cut the brass gear off? With a drillpress and bit, you might be able to thin the gear enough to crack it in half by twisting a screwdriver to pry it apart. Once the gear is gone the axle should remove much easier.
Tape or thick paper on sockets (tool) for wheel protection, or a hard wood with a stepped hole, can be used as cups for installing the new parts. Keeping all parts aligned well during reassembly is very important. It is easy to bend an axle while pressing.The result of a bent axle will be a wheel and / or a body wobble as your train runs, and / or rod issues and / or short gear life or gear bind (locks up).
Best of luck to you, and please let us know your progress.
