In general, yes BA will happily increase award inventory late on if a flight is not going to sell out however here's no hard and fast rule.

It all depends on the value they place on last minute seats on a particular route.

It is more likely at the moment as current BA strategy is 'bums on seats' taking whatever revenue they can. Award bookings generate decent revenue with either the fuel surcharges or the £50 change fee for a last minute MFU.

Making award seats available in CW when only expensive tickets are available probably reflects demand elasticity. If you are going to pay to fly CW at short notice you're probably going to pay whatever the cost because you have to go. Late award inventory may additionally attract some last minute 'what the hell, lets go to New York this weekend' travellers. This gives BA additional revenue without trashing the value of a seat booked at late notice.

edited to add....
Don't forget award bookings are (generally) free to change so you can book a flight that is close to the one you want and change it if the other one opens up. If using this strategy on the return ticket it may be better booking two one-ways as I'm not sure you can change an award booking once the outbound is flown.