Just from my own experience having also been through an "algorithmic" Google penalty, it took 6-8 months after uploading a disavow file to see any recovery. But good news is it does work. I agree with the responses from William and LJD but wanted to mention Google does not remove the links you disavow. That will have to be done by submitting requests to the website webmasters. Actually Google wants you to do this first before disavow, but it doesn't seem to hurt not to or do them at the same time.
I used RMOOV as a tool to do this. Great tool and very helpful folks who will talk to you on the phone! Although disavow may get you there with Google, I think it is worth the effort to remove as many bad links as possible since it just looks bad and other search engines will not know about the Google disavow. It is a lengthy process but tools like RMOOV and Detox can help.
So no reconsideration request is required unless you have a manual action. Going forward I keep looking at my link profile using OSE, GWT and Majestic and when I see something I don't like, I update the disavow file. Important to "append" that file so as to keep all the original links and just add on the new ones.