Not cycling your tank just makes it everything more stressful for you and for the axolotl.
Yes, tapwater is perfectly okay to use as long you as dechlorinate it if necessary.
If you don't cycle the tank, you're letting the axolotl sit in its own waste day after day. Even if you clean it out with a turkey baster daily (which you should) - some will still dissolve in the water.
Cycling your tank gives you a colony of beneficial bacteria that helps remove this waste that is
toxic and deadly to the axolotl. Doing this, you only need to do 20% change a week depending on the water volume you're working with and how much waste is produced.
Unless you're planning on doing tubs and daily 100% water changes, it's irresponsible and abusive not to cycle your tank. Check out the "sick" forum and otherwise - plenty of threads about injured, dying, or dead axolotls because people thought it wasn't necessary to run a filtered and cycled tank.
It's okay to put an axolotl in a small container and do 100% water changes with dechlorinated water on a daily basis because although you have no beneficial bacteria, you're taking care of it daily and a sick axolotl in the fridge has a slowed metabolism and isn't going to produce waste at the rate a healthy and comparatively warmer one is.
Check out ANY reputable fish forum. Cycling isn't just a "thing" people here do. It's a thing that's necessary and important to anyone keeping an aquarium with live creatures in it who don't want the stress of doing daily 100% water changes. Daily 100% water changes should only be done if you have a good reason - such as being a breeder and having your stock in many, many tubs - breeding is a LOT of work.
Extra chemicals like Prime are, unfortunately, not recommended. Axolotls do not benefit from "stress coat" products and bottled products are not any good at removing harmful chemicals from your water. Like another user said, it's like putting a band-aid on a broken leg. You're actually making more problems for yourself in the long run. You're preventing your tank from cycling, and if it *is* cycled and you're using those products, you're just throwing money away and potentially adding harmful chemicals to your water.
The best way to cycle a tank is with "fishless cycling" - using a bottled ammonia product to cycle. While you can cycle with fish or the axolotl - you can also kill them that way. Part of cycling is an ammonia spike - and if you don't have a test kit and you're not paying the utmost attention - this can kill them. Axolotls are especially sensitive since they have soft skin that is easily permeated by chemicals, unlike fish. This is another reason why many fish chemical products are not suitable for axoltols.
Sure, cycling can take weeks. But would you rather spend that time or do 100% water changes every day ? Because if you don't do either, you'll be back here with issues of soiled water, ammonia burns, poor health or death of the axolotl. Spending the time to properly cycle your tank saves you time, money, and stress - and the lives of whatever's in the tank.
And tap water, unless there's something very wrong with the local water, is always okay as long as you use a dechlorinating product as necessary. It's a good idea to have your water tested before making a choice. The vast majority of people use treated tape water, but it is always good to be safe and know what your "baseline" is - for instance, if ammonia is present in your water at baseline.