Postponement of flight due to illness

[BllGrhm]

My daughters husband was ill during a trip to Athens. She contacted BA with whom they had return tickets to the UK and was told by phone by two BA reps in UK to get a letter from the Doctor and when they turned up for the re-arranged flight to present this document.They were told they would not need to pay for the re-arranged tickets.
When they arrived for the re-arranged flight they were told at Athens airport they could not transfer flights like that.
With two young children they had to pay for the new flights and travel back. On their return they are getting nowhere with their complaint. I have tried on their behalf (the sum involved is over 1000 GBP). E-mails from BA just do not answer any questions. Has anyone found out the best way to proceed with a complaint like this with BA? Is there an independent arbiter. Do you take them to court?