Well it seems that the new London Overground management have got the same refund process as the other Transport for London services.
you can find the refund page here http://www.tfl.gov.uk/tickets/refunds/1092.aspx
The rules are for Tube refunds Tube journey has been delayed for more than 15 minutes by circumstances within TFL control, use the Tube refund link to claim your refund.
The rules are for journey on London Overground has been delayed for more than 30 minutes by circumstances within TFL control, use the Overground refund link to claim your refund.
You can read the Overground Passenger Charter one the link http://www.tfl.gov.uk/assets/downloads/Overground-Passenger-Charter.pdf
My claim was number on their site was #123
