I am afraid the bean counting w&*$ers in the MoD do this kind of thing all to often. It does not compare in any way but I had a vaguely similar experience when I left the Royal Navy.
I had to give 12 months notice to leave, yes a year to sort my pay out, and went to my UPO six weeks before going on terminal to check up they had no lose ends. "No its all in hand Sir".
So imagine my surprise six weeks after becoming a civilian again when I get a letter from MoD Accounts saying I owe the £19 due to "receipt of belated information". I was livid - seven years service and they have the cheek to chase me up for £19, but being sensible I wrote the cheque. Well I had also just received the standard letter the 2nd Sea Lord sends to all Officers on leaving. So I sent a reply to him which went on the lines off "when I was an Officer of the Watch there were no excuses for having the ship late at an RV, regardless of what belated information I may be in receipt off". He agreed, and investigated the matter personally on my behalf. I will not post his words but lets suffice it didnt take a degree in astro-physics to work out he had given a number of people in MoD Accounts and the Supply Branch of the RN a big kick up the behind.
So my advice to the poor lady concerned is refuse to give the money back, take it to very top and get it properly sorted. And make sure that those responsible are held to account.