Awhile back, I achieved Silver mid-trip and was denied lounge access at LHR, which I found completely reasonable. My boarding pass didn't say Silver, I had no Silver card, and I was traveling in WT+. But, it couldn't hurt to bring a print-out of your Exec Club account overview and state your case to the lounge staff, although this isn't proof of anything as such documents can be easily forged. If you're lucky, they'll let you in.

Do bear in mind that BA aren't obligated to do so if you have nothing official from BA with both your name and that of a precious metal on it.