I successfully was able to get re-booked by Qatar onto British Airways this morning for both my wife and me. Posting summary here in case helpful for anyone else.
Original booking: MIA-DOH-MLE, departing March 5. My ticket booked with points on JetBlue. My wife's ticket booked with cash on American. Economy class with paid seat selections via Qatar website.
New booking: MIA-LHR-MLE, departing today (one day sooner), fare class Y on both tickets. Ticket change was done by a 'senior agent' at Qatar and he managed both my JB and my wife's AA tickets concurrently. It took a total of about 1.5 hours to get this done once we reached him.
More background: we had been calling JetBlue and Qatar and American since Saturday trying to get tickets changed to BA with no luck. Effectively they all told us that until the flight was cancelled, no one could take any action. This morning I woke up to the news of Qatar cancelling through the 6th so immediately got on the line with JetBlue figuring I may have more luck getting the change done with them than Qatar. Long story short for JB: after over an hour, they told me they couldn't do anything and to call Qatar since they were in control of the ticket. So I phoned Qatar and got nowhere for the first 30 minutes, finally being told by the initial agent that he could only book us on another Qatar itinerary during the next 14 days. My wife then said some sort of magical, abacadabra like phrase that my company's corporate travel agent suggested and that helped us immediately. She stated something along the lines of 'A supervisor at JetBlue's interline agency in Orlando told us to ask Qatar to take control of the ticket and re-ticket us on a partner airline with available seats'. Something in that approximate wording triggered the Qatar agent to transfer us to the 'senior agent' who, within 5 minutes of explaining the situation and then specifying which BA itinerary appeared to have open seats and confirming that we could fly one day sooner (today), very affably suggested that we pack our bags and get ready for our trip. Then during the next ~1.5 hours he got us re-ticketed.
Fingers crossed for getting there and we'll worry about the return MLE-DOH-MIA closer to the date (17th).