How long does it take to apply for a dependent visa in Indonesia alongside my own visa application?

Usually the same amount of time. Applications are just sitting waiting to be reviewed. Initial applicant might be longer if they come back and ask for more evidence of work/soft power

Depends on how much they have to do right now. The reason it normally takes times is because its a huge load of papers for many applications, and many people dont really make their application clean and easy to read.

So depending on your application, how much paperwork it was, and how much they have to do when you apply for yourself and your kid. But you need to get your application approved first.