How long do I have to respond once my wife files for divorce?

I'm assuming that your wife has your current address, so that when she does file, you'll be served with her complaint.  With the complaint, you'll get a summons, which will give you the details on your time to file a response, and that time runs from when you are actually served.

The only way a divorce would be filed as "uncontested" would be if there were a written settlement agreement, resolving all the issues about money, property and children (you can't make a contract for the actual divorce itself).  You'd still have the right to contest the grounds for divorce, and if you had a good reason for wanting the agreement thrown out, you could make that part of the case as well.

Usually, a divorce is called "uncontested" because the defendant never filed an answer.