It's not actually a "first come first serve" argument as I understand it - it's that contractually the UK and EU supply chains are separate, each contract only requires them to supply the contracted vaccines from that supply chain, and the EU one is in much worse shape yields-wise because the EU dragged their feet on signing the contract for internal political reasons leaving AZ with less time to fix the problems. (There's a contractual provision in the EU contract allowing them to supply some vaccines from the UK which the EU is trying to stretch into them requiring it.) The whole claim that AZ is doing first come, first serve and it isn't fair seems to be EU spin, and pitting the EU against the UK seems to be an intentional EU tactic. In particular, as I recall EU officials started making evidence-free claims early on that AZ had been exporting doses from their European facilities to the UK and this caused the shortage, something they deny and that no evidence has been found for despite raids on their facilities.