I just received this reply from BA, I asked them what time the lounge opened, so I am even more confused now:

Thank you for writing to the Club about lounge facility.

It isn't possible to pay for access into our Executive Club and premium travel lounges.

To ensure that the lounges offer the most relaxing experience possible, while accommodating such a large number of people, access is restricted to customers travelling First, Club World or Club Europe or who are Gold or Silver Executive Club members. Unfortunately this means that we are unable to extend invitations into the lounges for customers who are not travelling in First, Club World or Club Europe.

I have checked our records and found that you are booked in (First class) cabin. Thus, I am sorry to have to disappoint you that you will be unable to access our lounge.

You can also find detailed information about access to airport lounges at:

http://www.britishairways.com/travel/lounges/public/en_

Please let us know if we can help you with anything else.