You have had better luck than me. I've never got beyond your first line, despite innumerable calls since cancellation on March 24th so I sent a section 75 request to Barclaycard yesterday.

I had three flights from EDI to JFK, booked for 1st/15th April. When BA finally cancelled the flights I went on the website and was only offered a voucher.

Called the Newcastle call centre and, after an hour on hold, got hold of a very nice lady who told me that a voucher was the only option. I told her that I was legally entitled to a refund as they can't fulfil the contract - she put me on hold to speak to a supervisor, came back 10 minutes later and said they would process a refund.
It took around three weeks for the money to hit my account, and I was relieved when it did.

I honestly don't think the call centre agent I was talking to was lying, or deliberately trying to leave me on hold to see if I would go away - she seemed genuinely unsure of what the options were, which seems craxy.

So all came through but only after a lot of persistence.

The money appeared back on my Barclaycard this morning .. curiously it has a date of 30 April, which was the day after I asked for it, but I was only notified today and the entry is BA refund not repayment by BC.

Not going to enquire to closely, getting the money is all that counts.

I had two bookings cancelled by BA. First was a "holiday" (car+flight) where I had only paid the deposit. 2nd was Avios + Amex companion voucher + taxes in cash.
In both cases it tooks several goes to get through but in both cases handled well when I did speak to someone and the cash, despite being warned "could be 6 weeks" was back on my card in under 2 weeks and the Avios and voucher reappeared during the phone call!
On the 2nd call I was told all the contact centre staff are currently working from home.

I am not surprised it takes this long. My brother is working from home, taking calls over the web and handling client emails - he says he takes very much longer to process each enquiry and his work is not related to airlines. As I said before, the airlines are not geared up for this and patience seems to win.