Back-end systems (refunds and miles) are the low points (in my opinion)
Even after all the recent cutbacks, bad financial news, etc., I still find flying on BA to be a pleasant experience, with cabin crew being the absolute high-point. Flying in the last few days after news of losses and layoffs, every member of ground staff and cabin crew I encountered was helpful and ready to be of service.
My complaint is about the back-end systems (if that is the right term to use). Getting a refund on refundable tickets is a painful, painful experience. It requires multiple phone calls, and e-mails, and sometimes even complaints to the credit card company.
Getting miles re-credited from cancelled award tickets is equally painful -- apparently a "new" system has been implemented and it now takes "7-10" days to get miles re-credited. Well, I requested re-deposit of miles over two weeks ago for one ticket, and I am still waiting. The "old" system gave refunds of miles pretty much immediately.
Also, WHY can't they fix the MFU mileage credit problem? It is such a royal pain to have to call and explain to the phone agents that you flew CW on a paid WT+ ticket that earns points and miles. The system works wonderfully when you aren't flying on a MFU ticket with pretty much immediate crediting of miles and points, but is worthless for MFU tickets.
Flying with BA is today still a good experience. But for me personally, the most stressful part flying BA is not the airport/ in-flight experience, but having to deal with the airline before and after the actual day of flight.