You will want to be absolutely sure that your mini-cycle is not going to show itself prior to adding fish, but if you've had a week of double-zeros (zero ppm ammonia and zero ppm nitrite(NO2)) then you'd probably be fine to begin your introductions. I'd start by bringing your neon shoal up to minimal standards. You pretty much never want a neon shoal to drop below 6 and in your tank (without looking back at tank size) I'm thinking you could easily go 8 or 10. Not only will this help the neons relax and be more healthy and less likely to nip the other fish, it will also get them in there and established prior to the angel introductions. You have the correct idea, get small angels and introduce them after the neons and they will, at least for a very long time, consider the neons to be part of the scenery and not something to eat. Be aware however that angels are among any number of cichlids (and gouramis for that matter) which have varied personalities on an -individual- basis. The very thing that makes these fish so endearing to their owners, the individual "personality" aspects, can mean that some individuals are, at the extreme, either mean or very nice to other particular fish. Note that if it eventually does turn out to be a problem, it can be hard to spot at first. Angels eat neons as their basic food in the Amazon and what they do is glide and sit pretty still and then make a very rapid move and grab the neon whole. The shoals of neons they eat from can number in the thousands. Still, this is just to understand. I and many others have kept angels and neons for their full fish lifetimes without incident.
Hatchets might also be a problem nipping the angel fins but you'll just have to see. They too might do better in larger numbers but I would understand if you end up pushing up against your stocking ceiling (one inch of what each species could become when adult mature per US gallon not including fins, for the first year ideally, until you become much more comfortable with the maintenance, your particular tank chemistry and your particular fish personalities.) So basically, between the 3, the angels, the neons and the hatchets, you have 3 fish that could all have some potential to nip each other but most likely will not and having the neons especially and the hatchets to some extent in larger shoals would be one of the main reducers of the nipping possibilities, whereas some other aspects, such as the angel personalities, would be unpredictable but not something that should stop you I think.
~~waterdrop~~