Others may have more personal experience with that particular route, but it's important to understand how and when redemption seats are released, as it's certainly different to how many US carriers operate and has, anecdotally, been subject to change recently.
It works something like this: BA may release some seats when the flight is loaded (c 355 days out) and then at any time up until departure, depending on commercial loads, They certainly won't look to fill all the 'empty' seats (to maintain the exclusivity of the product ?).
So, it pays to keep checking a couple of times a day right up until 24 hours out.
Also, upgrade and full redemption seats come from the same inventory, so you can easily check online for a full redemption and call to upgrade if MMB doesn't want to play.
Further, availability can dry up completely during sale periods and reappear later.
Make sense ?
hth