Ill start by saying i have little knowledge of axolotls except that they are cute.
Cycling a tank typically takes 6 to 8 weeks so you have only just started on your fishless (axolotless?) cycle.
A fishless cycle starts off at 4ppm ammonia and you redose to 2ppm when it drops below 1ppm. This stage can take a few weeks. You then keep redosing 2ppm every time it drops below 1ppm. When you are able to dose 2ppm and 24 hours later you see 0ppm ammonia and nitrite you are cycled. Do a big water change to reduce nitrate and add fish.
The best way to speed up your cycle is to introduce some filter media from an established filter. Perhaps you have a friend who keeps fish who could let you have some? Another option is to use a bottled bacteria product like Dr Tims One and Only. This might help but these products are hit and miss whether they work or not. At best they will speed up the cycling process to weeks rather than months. Increasing the water temperature to 29/30c will help also.
Did you do anything wrong by doing a water change? Not really. You have reduced the food source available to any growing beneficial bacteria but a water change is often done to kick start a stalled cycle, so no harm done. The only thing i would say is you didnt really give it enough time, especially if you did nothing to speed it along.
As to your nitrate levels, my understanding is that axolotls arent that sensitive to nitrate, so it should be possible to keep them around the 40 ppm
area. My short research says they are ok upto 60ppm. Someone might correct me here. Nitrate really isnt that much of an issue as long as you don't let it get totally out of control. If you wanted to lower the nitrate levels in your tap water consider filtered water, either a filter on your tapwater or buying filtered water.