Thought I'd better update this even if I am a year late...
The flight from Madrid left about 20mins late but we still arrived at LCY on time. Immigration was a breeze as I was the only person in the non-EU queue. I was also the only one with bags to collect but they came out after only a few minutes of waiting.
The DLR was right outside and I was able to use my AMEX card to beep on. A train arrived after 2mins.
The transfer to the Jubilee line at Canning Town was literally out the door and down 1 small flight of steps. The train arrived almost immediately.
The transfer at Green Park to the Piccadilly line was a bit more of a hike then I was expecting. Lugged my Bag, Backpack and a Moroccan Carpet up 2 flights of stairs and then about 300m through a hot tunnel.
But once I was on the Piccadilly line I knew I was pretty much home free barring a train breakdown. It was pretty crowded on the trains as it was peak hour but I manged to wedge in a corner and not inconvenience too many people. Once the train cleared Hammersmith it pretty much emptied out and I had a whole carriage to myself.
We cruised out to Heathrow where the final bit of pain was the trek from the station up to Qantas First Class check in. From then on it was all sweetness and light. And champagne. Lots of Champagne
So in summary it took me about 1.5hrs to make the transfer from LCY to LHR and is no real impediment to taking an Award ticket that includes this tranfer.