Barry, I was due to fly with Emirates last Sunday, to our long-awaited romantic honeymoon-like holiday - last opportunity before our baby arrives... When we learned that the flight was cancelled, this was on Saturday, we immediately called them. No getting through to Emirates UK, but their office in Dubai is open 24h and we got through to them easily enough. We asked to be re-scheduled as soon as possible, but they offered us seats for Thursday/Friday at the earliest - although we had seen online that they were still selling cheap (!) seats for Wednesday. When we asked why we couldn't have those, they said it was because they were in the wrong booking class. They could only re-schedule us within the same booking class (that's not economy vs. business vs. first - rather, within economy, there are numerous different booking classes, often indicated by letters, eg. "F"). We asked if we could pay extra, and one person said yes, for £300 per person (that's more than the entire price of that flight leg); but when we called again to do this, another person on the line, said they can't do that. We ended up asking for a refund for the cancelled flight and booking a new one for Wednesday (tomorrow) online... the new one actually worked out cheaper than our original, cancelled flight!
Of course, no-one knows if it will actually take off on Wednesday...
This is just to say: Call Emirates, if necessarily directly in Dubai; ask about possibilities if you were prepared to change booking class/pay extra; if the answer is unsatisfactory, call again a few minutes later, it seems you can get lucky with different persons on the line! But you really must let them know if/that you are prepared to pay extra, else they will ONLY check for rescheduling-availabilities at the same price/same booking class - no wonder there is nothing available! And if you see a reasonably-priced flight online, just book it; and then get the refund for the cancelled flight.
Good luck!