If the ammoina spiked and dropped , you are fine.
Diatoms , algae are not an indication of the nitrogen cycle.
What we need is the bacteria to eat and absorb the ammoina that fish will produce and that is produced when foods rot.
Nitrate and nitrites are just an indication of the process. Neither are toxic to a fish.
So imo. Your tank is in fact cycled , so go get some fish.
Try not to overfeed , as the biofilter is not processing the nutrients well, so they build up and feed the algaes cyanos etc that are associated wit the ugly phase of the cycle.
Fwiw, Diatoms need N/p and also a lot of silicates, usually supplied by dissolving of the outer layers of dry rock. Once the silicates are close to exhausted , the diatoms fade as they no longer have the minerals needed to form their skeletons. This is why diatoms feel different between your fingers than cyano. Diatoms have bones.