Sometime yes, sometimes no. I was the only person in a First cabin from Barbados, so thinking I would get some really attentive service.
At about 24 hours beforehand, they opened up the other 7 seats for Avios, and the First cabin was duly filled with Club World customers upgrading via Avios.
But there are other times when they stubbornly refuse to open the availability. It all comes down to what BA revenue management consider the likelihood of being able to sell a last minute juicy fare, which will be route/date dependent.