[Only just seen this thread...]
Domestic UK flight for May 2020 was cancelled on 9 April 2020. We started ringing the advertised UK phone number on 21 April, at least twice a night. "It's a difficult time, we're very busy", disconnect.
On 24 April we finally got through to a human, who did sound quite stressed, but who started the process, saying that it would take 3 weeks.
On 29 June, with no refund, we started ringing again, to get the same "It's a difficult time" message and disconnection.
Came across the number for BA's Gold Executive Club members (+44 800 123 111). Rang that, got onto someone, explained the problem and apologised. They were sympathetic, checked and said that we were still in the queue, but now escalated.
Finally, the refunds came through to the credit card on 18 July. Overall, 12 and a bit weeks. Thankfully phone calls to the UK are now so cheap ... at the old rate of $3 a minute, we would have spent our refund trying to get our refund.<g>