I don't think it is as cut and dry as that. I have never signed for the aeroplane in my life and who signs the book in the clubhouse is whoever gets there first........
I know UK pilots like cut and dry cases but in the real world, it is never like that. Imagine another scenario - Aeroplane enters IMC. Pilot A (LHS) does not have an IR but pilot B (RHS) does.
Are people really saying that Pilot B should sit back, and relax as the aeroplane plummets to the ground because he cannot become PIC in the air? I don't think so......
