The target used to be ten working days. Seriously, it is only if the court is in absolute crisis would it take six weeks, assuming the petition was filled in correctly. Yes, he would receive notification.
Do you know which court it is. They will be able to tell you how long that particular court's backlog is if you ring in, though they won't be able to tell you about the divorce itself. Of course, if you do that and find him out it will he will blame you and make it your fault because you should trust him. He will tell you that it's because you don't trust him that he hasn't filed. If he loved you he would file.
Honestly, stuff with a fee usually has a turnaround of a few days at most under normal circumstances.
You could be kidded along with, 'I forgot to sign it, so they returned it and I sent it back... I used the wrong form so they returned it and I sent it back... I didn't put the right addresses in so they returned it and I sent it back... I forgot about the arrangements for children so they returned it and I sent it back...' etc etc etc.
If he wanted the divorce he would already be divorced. If he wanted you in his life then you would already be in his life. Sorry.