Have you tested your main tank??? When you mixed 40% main tank water with tap water I would expect to see 0,0 on ammonia and nitrites unless some were coming from your main tank/tap water. (more likely to be your tank tbh).
I'm more worried about your main tank and fish than about this nursery tank. As those first set of results imply you main tank was at ammonia 0.8 and nitrites at 0.5
Good points, the levels have dropped in your nursery tank meaning that there are some bacteria in the filter

Unless you are going to feed the bacs with some sort of artificial ammonia then wait till the levels drop to 0,0 then introduce fry or an adult fish to keep the tank cycled.
Problem is you wont know exactly what amount of ammonia the tank can cope with. So even once you've got 0,0 results you may find your levels increase again once you introduce fry/an adult.
Eitherways, don't do any water changes and don't just let the tank sit once your levels drop to 0,0. If you just leave it then the filter bacs will begin to die off and you'll have to cycle it all over again.
Do you have fry waiting to go in it already?
Oh and the Nitrate 'spike' you refer to isn't a spike at all.
It's the result of the filter bacs completing the nitrogen cycle and so is a good sign.
However I would imagine your previous reading of 2 was probably false...or you left the 0 off and it was meant to say 20?