The purpose of seating an adult near an adult is safety and not convenience. The former is cost-free. The latter is a consideration in purchasing tickets (and paying a fee).
While the entire notion of paying for seat assignments may be abhorent, it is no more or less abhorent for a family than for an individual and it is hard to justify dislodging someone who has an assigned seat for the convenience of one who does not. How would you like to be the person who gets punted 10 rows back for other than a safety concern?
The laptop thing sounds as if BA simply got it wrong. But, it did not really matter as you checked it for IST-LHR as you were required to and you have already explained that you were not in a position to reclaim your bag at LHR and recheck it at LHR.