BA dont charge a processing fee to refund your taxes - even on the most restrictive tickets.

As others have said:
1) There might not be much left given most taxes relate to the outbound UK leg
2) No Show doesn't make a difference. I missed my flight recently, i went to the ticket desk to try and rebook and was quoted an absurd fee to change. I said no - please cancel and refund the taxes - they said this had to be done through the agent that sold the ticket (ie BA Exec Club). I didn't get round to calling them to arrange the refund till a couple of days later.