I don’t see how this would work without payment up front and frankly it is not a big deal for me if they are full as it is not pleasant (when BA was using it as a fallback and it was rammed then I frequently chose Pret as a better option). There is always the MyLounge as an alternative which never seems to be busy.

Edited to add that you have to pay a £5 fee upfront to reserve a place via their website if you are PP or LoungeKey. Not worth it IMO.