I imagine any hardware that is running an OS is going to take more time to power up that you need.
If the Omega had a sleep mode that'd be the way to do it but as yet, that's not an option.
What is the reason you need such quick power up? Is there some other way to tackle the requirement?
Cheers,
Chris.
The 'Magic Smoke' is what makes it work! If you let that out, no workee end of story 
@crispyoz
It's tricky to find any data on the current sourcing capability of the GPIO pins. I only had a quick look in the Onion docs and the Mediatek datasheet without success.
You're drawing ~15mA from the port with the arrangement you describe assuming the 1.8V LED drop per @Neil-Manuel's reply. Once you find the maximum current source rating on the GPIO pin you can make a call.
It's also worth checking the sink current rating vs the source current as sometimes the sink current is the greater amount. In this case, you reverse the LED(s) direction, attach the resistor(s) to +Ve and switch the GPIO off to activate it.
HTH.