All their hotlines in Europe are closed and they do not have any 24h hotline.
I have even tried to call the Canadian and American hotlines, which were still open, but my calls were rejected because they were full. I also tried to call Iberia (because the return leg was on Iberia) but they could not do anything.

Now I think my booking is lost and I will have to cancel my whole holidays just because of BA website bugs ... I will still try to call BA tomorrow to make them understand how catastrophic their support service is.