It’s been said, seat allocation is not the same as having a reservation. Check you have a ticket number 125-#. TomHoward and others are correct, BA only allow certain seats to be bookable depending on both function (e.g. bassinet seats) and frequent flyer status (depending on aircraft, long-haul or short, and fare class). For example if you aren’t gold you can’t choose the first row on short haul until near check in. And the cost of seat reservation changes by frequent flyer status.
If your flight is a few months away it’s likely the schedule or aircraft type will change anyway. I would not worry about seat selection and just check regularly in case seats open up, and at 72 hours and at 24 hours before the flight. You can use the free ExpertFlyer profile to check which seats are actually reserved versus held for operational / status reasons for your particular flight.
Absolute worse case, if you were chosen to be downgraded at the airport you are entitled to 75% of the fare in compensation (for that leg) or you can ask to be rerouted.