BA in India are very aware of this, and sometimes have staff outside airports helping people to get in. Allegedly they are trying to get a second form of check-in document for India departures but there's a backlog...

As ever, the easiest way out of this, so long as your correct email address is on the booking, is to use the Finnair email link. This gives an email with a barcode on it, and a PDF, also with a barcode. This works very well indeed.
https://www.finnair.com/gb-en/custom...ticket-receipt

Also keep trying the old URL, it sometimes works after a few hours.
First try the Congo entry point
https://www.britishairways.com/trave...s/public/en_cg

then try
https://www.britishairways.com/trave..._gb?eId=104509