They will never sell or run a First service at short notice, it's just not what BA does, and BNA isn't the sort of route to go First. Instead, there is the option of Flub - where some Club passengers will be placed in the First cabin, but receive CW service. There is a very detailed thread on Flub with lots of data points, it happens quite a lot. Gold cardholders tend to get the Flub treatment. But if BA does not need to use Flub they won't, since there are adverse staffing implications that stem from using the First cabin.