You haven't stated in which category this petition was filed, i.e. whether it is a U.S. citizen filing for a child or a U.S. legal permanent resident filing for a child.
That answer will of course affect how long it takes for visas to be issued, since there is a backlog in the latter category (that cannot be fixed through an expedite).
Let's assume you are U.S. citizens and that the visas are therefore immediately available. The expedite that USCIS granted you in the processing of the petition itself does not carry over to the State Department (except in a few limited cases where the State Department takes it upon themselves to expedite a case without being asked). So you will need to separately ask them in your correspondence for a new expedite.
A non-expedited case where the visas are available can progress through the National Visa Center in two to three months IF you attend very closely to the documents requested. How soon the interview is set up at the consulate will also be a function of how backlogged that particular consulate is. You haven't told us which one it is.
Without this information, I would guess that the average case from a do-it-yourselfer paying close attention to documentation spends three months at NVC and another two months to be set up for interview.
Assuming that you are talking about immediate relative children or stepchildren of a US citizen who are not subject to quotas, you need to submit the DS-230 applications and other original documentation to the National Visa Center. You say that you have paid the affidavit of support fees, but I assume that you have also paid the other immigrant visa fees. If so, then you need to submit the DS-230s and documentation required for immigrant visa processing (which varies depending on the home country of the applicant, age of the applicant, and relationship to the petitioner). Once you have submitted everything to the NVC, the soonest the children will be scheduled for the visa interview at the consulate is one or two months later. Once the NVC has scheduled the interview and sent the file to the consulate, then you will need to contact the consulate directly to request an earlier interview date if there is a reason to expedite.