I have also experienced BA's nightmare impossible online system to claim for damaged baggage. You have to get a Report number (called a DPR number, 5 letters & 5 digits) BEFORE you can successfully open a claim online.
But to try an get the essential report number (if you have not got it at the airport), is impossible on their website or by calling their published numbers - their website is set up for circular process - you go to the 'Making Baggage Claims' page where you learn you have to first get a Report Number, that page has a Contact Us link that takes you to the contact options, and the only relevant option there for baggage claims takes you back to the first page! Even if you find the page with the link to raise a baggage report, clicking the 'Report' button again takes you back to the 'Making Baggage Claims' page.
Its absolutely shocking.
Neither can you get through to a person on any phone numbers - their automated system just send you a text with a link to the 'Making Baggage Claims' page.

HOWEVER - the number for their baggage team in the UK is 0344 493 0785 to get the Report Number / DPR you MUST have to raise a claim, and from a real person!
HOWEVER, when you call, it still initailly goes to the automated bit to state why you are calling - DO NOT mention 'baggage claims' or the word 'baggage' or 'bags' to the auto system, or you just get the text and the call cut off.
I found say something like "to get a report number" seemed to get me through to a real person. They will ask for your file number - just explain that you need to get a 'Report Number (DPR number)" for a baggage claim. Have your bags tag number, booking ref etc. to hand. They will ask various details about the bag, value, damage, and your details and open the claim and tell you he DPR number.

Then you go online and scroll to the bottom of this page (britishairways.com/content/information/baggage-essentials/lost-and-damaged-baggage/making-baggage-claims ) and click 'make a claim' and fill in the info on the 8 pages and submit.
NOTE: i put in the correct booking ref and last name and it didnt find it, but if I logged in to BA.com before opening the claim, then there was an option on that page to select the flight from your past bookings, which then worked.
You get a claim number - they say they email it to you also but they still havent emailed mine, so MAKE SURE YOU WRITE IT DOWN!

I hope this helps others to raise claims for damaged baggage.

BA has become a toal customer service joke TBH - near impossible to speak to anyone, and their website is so messed up that processes dont work or are circular. Its s shame their IT management is clearly incompetant that they have these issues. BA should loose their licences to operate at all TBH.