Adam,
Welcome to FT and the EK forum.
You can check award availability on
www.seatcounter.com. Its semi-accurate, but should never be depended on.
Here's a screenshot of the 13 August.
Most airlines refer to Business Class by the codes C, J, D, Z or I.
The common thinking on this forum is that EK pull award inventory out of the "D" bucket, and you'll see by the screenshot that there are no D seats showing.
Doesn't mean that upgrades aren't available - it just means that for whatever reason, EK's Inventory Management (IM) system is holding them back.
I actually made a dummy booking and out of the 42 seats in business, 24 were showing already sold, with 18 available.
I think this might come down to the gate, but you then going to fall behind any Gold's who are wanting upgrades as well. In addition, expect EK to sell more seats between now and departure.
I'd keep calling if I was you. The golden rule to obtaining upgrades is to be persistent, especially with EK.
Good luck.