The calls should be recorded, but in any case it's up to BA to disprove your friend's case, not for him or her to prove

However, their rights here are somewhat diminished given - as has been said - BA didn't actually have any legal obligation to move you over to EDI under UK261 law. That said, having been assured your friend had a reservation, they might well be entitled to compensation under Article 4 for being involuntarily denied boarding due to BA's failure to ticket. Obviously they're also entitled to a refund as the delay was over 5 hours.

Lastly, you personally don't have any recourse as you travelled (other than getting the Avios difference back between F & J). Your friend needs to make any complaint/claim.

This feels, to me, a better case for CEDR rather than MCOL, given technically QR denied boarding, but others more expert than me can advise on that.