Once you have established your nitrifying bacteria, you should never see ammonia or nitrite, only nitrate and in some systems, especially ones with live plants, you may not even see that either.
In order to get a nitrite reading in a fully cycled tank, you need to put in more ammonia than the ammonia bacs can handle. You need to cause them to reproduce to handle it and that would mean the greater numbers would be producing nitrite faster than the nitrite bacs could convert it. And this would produce a nitrite reading. In a fully cycled tank there is a "proper" balance between the two bacterial colonies such that the nitrite bacs can handle as much nitrite as the ammonia bacs can produce.
There is one big difference between adding ammonia to a tank to get it cycled and the way ammonia is produced in that tank after it has been cycled and then fully stocked. In the case of cycling we add a full day's worth of ammonia which would result from full stocking all in one go. In a stocked tank it takes about a day to produce that total amount of ammonia. This is why in cycling one is told to add 2 or 3 ppm and then test in 24 hours. If you wanted to replicate while cycling how things would actually happen in a tank, you would be dosing very tiny amounts or ammonia every few minutes all day long that combined would equal that 2 or 3 ppm.
Once you have finished cycling the tank but have not stocked it, the bacteria need some ammonia to keep them going. So you dose an amount greater than what they can handle instantly. So you will have some ammonia reading while all that ammonia is waiting to be converted. But as it is converted the nitrite bacs will immediately do their part. Unlike the ammonia which enters the tank all in one go, the nitrite is created over time by the ammonia bacs. The nitrite bacs are able to handle that amount as it is produced.
The end result is you will be able to test ammonia in the tank from the time you add it until it has all been processed some hours later but you wont get any test results for nitrite. However, if you added 1 ppm of nitrite to the tank instead of ammonia, you would test that for some hours just as you could for the ammonia.