As far as I can tell the conditions for a BA refund via a UK claim seem to be these: you must have flown a long-haul flight on BA; the ticket on which you flew that flight must have been issued within the specified timeframe, whatever the actual date of the flight; the ticket must have been issued by or on behalf of BA (ie a 125 ticket); and the ticket must have been issued in the UK. I couldn't find all of this set out explicitly, so this comes with the caveat that it's my interpretation of the stuff on the website and may be wrong.

Even if it had been relevant to my claim, I wouldn't have trusted the system enough just to submit a BAEC number. If you do it by date of flight, the earliest possible flight would have been one on the start of the period, and the latest flight would have been one year after the end of the period. Fortunately, I still had all the ticket receipts.