itch.io

yeah both mean to repeat. It's also correct that nachsprechen refers to speech, but not when you repeat your own speech. Nachsprechen is used when you want someone to repeat something you say word by word. Examples would be when someone teaches a kid how to pronounce a word and says "sprich mir nach: [insert word here]". Another use is for example when making an oath. Someone will tell it to you line by line and you are supposed to repeat it 1:1. 
Wiederholen on the other hand is used when you repeat something you said yourself. The particular sentence structure you used is typical for announcements via speakers. Those are repeated once in case the listener didn't understand everything. You also use it when asking someone to repeat what they just said (because you didn't understand it).
So basically while both translate to the same in English, nachsprechen is way more specific than wiederholen.