Thanks again everyone for your input here; in the end Royal Mail were very prompt delivering (09:20) and it made an exit unnecessary.
Hopefully this will help the next person though!
OT: Seems BA aren't the only ones having lounge teething troubles, CX are serving a much reduced menu in their Flounge: only starters, desserts and items from the noodle bar. No main courses. Apparently it's due to an ongoing electrical issue in the kitchen.