My advice:
1. Go to CX CS in HKG, explain, and see if there's anything they can (and are willing to) do. You may be surprised.
2. If (1) fails, fly on CX's first flight to PEK. On arrival, go to BA CS, explain what's happened, beg forgiveness, and hope they can (and are willing to) help. If the next flight is the following day and they do allow you to move, claim a night's hotel on your insurance (see below).
3. If (2) fails, call your travel insurer and explain what's happened. Assuming you left reasonable time between flights originally then the 'missed departure due public transport failure' will kick in and they will help fix you.

Of course, and it should go without saying, for (1) and (2) above you are relying entirely on goodwill, so be the nicest, smiliest, friendliest customer you've ever been and keep your fingers crossed. Good luck.