So I know that おそい, ちこく, and おくれる all can mean late. But I don't know what context to use which in. Can I just swap おそくなります, ちこくする, and おくれる at will?
You can be "simply late" or "later than you are supposed to be".
遅くなります can be used in both cases but tilts towards the former. Both 遅刻する and 遅れる can only be used for the latter.
For example, すみません、遅れます/遅刻します is a definite admission of guilt, but すみません、遅くなります would most commonly be interpreted as "I'm not later than I'm supposed to be, I'm just later".