It should be fine. I have accessed GF a number of times while waiting for my replacement QF Platinum card. I only had an expired card and a printout from the online page showing my renewed status (I did not even have the printout on some occasions). The lounge 'dragons' did need to make a call on some of the occasions to check my status (other times they could apparently check online), and I felt rather bad about taking up their time, about which I apologised to them (along with the fact that I didn't have a new card), but all of them were very pleasant to me and let me in.

Just be very polite, non-demanding and explain what's happened and you should be fine. With your membership being BA, it should be easier for the dragons to check your status too than in my case.