Sorry for ressurecting an older thread but I have some recent experience with this.
In Sept 2023 I was stranded in GIB by BA when my return to UK flight was cx'd some 3 hrs before departure. No warning, no email, just happened to check the app prior to heading to the airport.
On the app I managed to rebook for the 1st flight the next day, but the airport were unhelpful with sourcing overnight accomodation.
As the OP said - no way to speak to BA. Customer phone numbers were country of departure specific, and in my case the number was in Spain and every time I rang it they just hung up without speaking to me (the UK number was non-geographic so unobtainable).
I posted on twitter asking for help, tagging BA and got a response from BA who helped me with my question (I was looking for guidance on the level of acceptable expenditure for a hotel given I had to book it myself - they said "upto about £200 would be considered reasonable" - I booked at the HIX for <£170)
Upon return to the UK, UK261 and expenses claim submitted without difficulty (I had screenshot the app showing the cx of the original flight), this was paid some 8 weeks later.