Well, the following is from my experience with aquariums more than axolotls, but should still be applicable...
If you can find tests for nitrite and nitrate, that should at least help determine if the biological media in the tank is developing and functioning properly. Since ammonia is the first step in the cycle, the presence of nitrites would indicate that ammonia is being processed. Nitrates would be a step further in showing the cycle is working. Often as the biological media develops, there will be first an ammonia spike, then a drop in tat at nitrites develop, and then eventually a drop in those as nitrates develop.
So if you can test for nitrites and nitrates, then that will at least let you know if the cycle is functioning. You'll definitely want to get an ammonia test if possible to help make sure that the levels decrease to a safe level.
In the mean time, and even after you get a kit and are sure the tank is cycled properly, doing regular water changes is a good way to keep the tank at safe levels. Until you can really test to be sure daily changes certainly won't hurt, and once you can test you'll still want to do regular water changes (weekly, etc) to help keep nitrates at a reasonable level and keep the water quality good.