"When we were doing self-introductions in turn, one student took [way] too much time and [he even spoke] with [very] bad pronunciation. My teacher seemed [he’d had enough and] wanted to move on to the next [student].”
You don’t need “himself” because self-introduction implies each person does it by him/herself.
Use “speak” because he was introducing himself in English, to him a foreign language. It’s "speak English", not "talk English". Talk means more of having a conversation. "We talked about the English language."
To have had enough means you don’t want any more of it or you can’t take it anymore. E.g., “I'm leaving you, Bill. I've had enough!”
>"次の人に進む" の感じは、"move on to the next" で、自然でしょうか？
Yes, that was good.
>"もう次の人に進みたがっていた" の "もう" のところは英語でどう表現すればいいでしょうか？
もう in general is already. “Are you going home already?” "Let's go! I want to get going already!" So you could also say “My teacher seemed he wanted to move on to the next student already.”
