There is a delay with my Denon AVR turning on when using my Roku Remote. While the LG TV will turn on immediately, it may take five to ten seconds for the AVR and sometimes, I'll need to use the Denon remote to separately power on the AVR.
This is normal behavior - your Roku remote is programmed to power on/off the TV, not the AVR - your AVR is being powered on by the LG TV (most likely via eARC/ARC CEC) after it powers on.
There is nothing you can do to speed this process up, aside from using a universal remote that powers on multiple devices with a single button press (e.g. Logitech Harmony), and even then there is an order of devices and delays.