I had my LBA flight cancelled in January, and was told at by BA agents at SFO and LHR, and by phone, that if I got the train I'd get my train ticket refunded. When I put in expenses for my (second class) train fare, lunch, and a taxi from my local train station to my home the customer relations team told they wouldn't pay the train fare (but would pay everything else), and would never have said that they would pay the fare. They were insistent that they would not pay for my train ticket, whatever I was told at the airport and by their agents. One MCOL later and we're now even (they settled before action was taken, so I withdrew my case, but they paid the MCOL fee).

So, if anyone told you that you could get the train paid for I'd stand your ground, but it seems like their policy is that they won't pay for your train. Good luck! MCOL was very easy to do, the hardest bit was summing up my case in only a few characters.