As someone who has missed a 06:20 LHR-MAD flight this summer due to turning off an alarm when unconscious and sleepy, BA said that there is nothing that they could do despite a very helpful agent at the First Wing. At least I was connecting to JFK from MAD so I was able to pay the no-show fee ( which was luckily USD 150 in my case ) and since it was not the first leg of the ticket, historical fares where in effect so, in the end, I arrived at JFK, 5 hours later than planned, however, got the Spanish taxes back thus I lost around USD 95 for my dummy mistake.

So, it's a YMMV situation. As a side note, I was Gold back then ( not GGL ) and since the ticket was at airport control, the Gold line ( got directed to India ) refused to touch the ticket as it was under airport control.