The API issue has been going on for awhile, and there have been a couple of threads about it. Maybe someone knows how to search for those, as there were several solutions suggested. One is to go through Manage My Booking without logging in to your BAEC account (if you have one). That worked for providing my husband’s API, which had caused problems when I went through “ my trips” while logged in (we are on the same ticket). Another solution that I saw here is to go through the Qantas booking system, using the same BA confirmation code/PNR. But I don’t remember the details on that.
I don’t know that the customer service agents can help even if you manage to get through on the phone. As for providing API at online check-in, I don’t know that you can count on that working either. Hopefully one of the experts will come along and confirm one way or the other. But FWIW, our last few flights with BA I have had trouble with on-line checkin even though all API was correctly entered for both of us, and even though we have been BAEC members and flying with them yearly since 2010. So I always pay to select our seats in advance.
Whose ticket—yours or your wife’s—-are the kids on? And how old are they? I believe BA has special provisions for seating for families with children under age 12, assuring that they will be seated with at least one parent. And maybe earlier check-in than 24 hours ahead?