I think OP meant no filter in the containers they are living in temporarily. They can live without a filter if you change the water every day like you are doing. Most cycles take a few months when done properly, and the only way you are going to be able to tell when it's done is to test the water with a master kit, do not assume after 9 days it will be done. It is possible to cycle the tank with the axolotls in there, and actually makes it faster because they produce a source of ammonia which the bacteria feed on. Just make sure you do 20% water changes every day to avoid it building up and harming the axolotls while it is still cycling. They would probably be happier in their tank too!
You can choose to cycle the filter with your animals in there, with temporary animals like goldfish, or with no animals at all by putting in an ammonia source like food. Some enthusiasts debate the ethics of exposing your animals to ammonia during the cycling process, but considering you have them in tubs with no cycled filter they would be exposed to a little ammonia anyway, about the same as if you were changing the water regularly in their actual tank... if not more because I’m assuming the containers are smaller.
I cycled my tank with my axolotls in there, and it took about 2 months.
Also, a live plant is very useful during cycling because it eats up some ammonia too. I grow a peace lily out the top of mine, but you can use java moss or ferns if you want a submerged plant as they can withstand the low light environment required for axolotls. Good luck