Not had the Start Stop warning, but from day one, I've noticed my GTD (from cold) takes a few seconds to turn over before firing. When it's warm it's instant.
I found that (as I have keyless), that pressing the start button (without pressing the brake pedal), switches on the ignition and hence the glow plugs, and then as soon as the needle sweep has finished, I then press the brake pedal re-press the start button and the car starts immediately, so don't know if if just needs a few seconds for the electronics to sort themselves out ?