First, my sympathy on the loss of your bag. A very annoying thing to have happen. I hope there was nothing overly valuable or of a sentimental nature which was lost.
You might find my Ten Commandments handy for future travel:
The first question I would ask is have you tried claiming on your travel insurance? Even if you didn’t explicitly take out a policy, you may find your household insurance or the credit used for the booking provides some limited cover in this instance.
Second, did you make sure BD (bmi) checked your bags all the way through to Brunei? Do you still have the bag tags with this ultimate destination written on them? If you do have that Brunei tag, then the above helpful posters are spot on; SQ is responsible for the loss as the ultimate handler irrespective of whether they received the bags from BD or not; they should take this matter up internally with BD but your claim remains with SQ.
Here is an entry on the Warsaw Convention which may be of use, including links to the inpenetrable full text:
If, however, BD only tagged your bags through to London, and you did not pick them up and transfer them on to your next flight, you would be liable, and it may well be worth checking with BD in London to see if they still have your bag (assuming this didn’t occur too long ago). I would not be surprised if this was the case as BD as getting quite tricky about through checking bags since they went all low cost on us.
I rarely trust an automated transfer of bags and wherever possible personally handle my bags on connections like this.
Good luck in obtaining some compensation for your loss. Do keep records of every contact you have with the airlines, and remain calm even though you must be exasperated with the way in which your situation is being passed from pillar to post.