On Mar 25, I was able to get through to British Airways to get a refund on a flight they canceled that was scheduled in April. The next day March 26 ( or 26 March for our European friends), the refund showed up on my credit card statement. That was for the airfare. I was told the seat fee would be handled separately.
I would check periodically but the seat fee refund never showed up so I I just wrote it off as I didn't want to wait on hold to speak to someone.
I just checked my statement and on June 3, the seat fee was refunded to my credit card. It only took a little over two months.
The moral of the story is......be patient. It may take awhile to see a refund.