http://www.britishairways.com/en-gb/information/family-travel/getting-ready-to-fly
This link has all the info about infants and children / family bookings
The info you got from TA is strictly not correct
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If you have an infant ( under 2 ) you can book seats on booking your flight
If you travel with children you can pay to sit together ( reserve) or BA will at 5 days out allocate seats for children together with an accompanying adult ( could be just 1 though ) !
You have already paid to reserve seats ( row 14 ) if you hadn't then you would have to wait to OLCI at -24 but as you have paid you can go into MMB anytime and change your choice , as at -72 all seat preferences that have to be held for families and status members have been picked , what's left over is available for people like you who have paid to reserve and want to change their choice or people like me with silver status ( bronze also can ) to pick any unreserved seats that are held back for these passengers , if no infants on flight then bassinet seats are available , if no golds then row 1 or exit seats they become available
No problems , good luck !!