Is there a way to speed up the delay when pressing the home button? My guess is that the reason there's a delay when pressing the home button is due to the fact that it could be re-purposed into launching "Google Now" or "S Voice" by pressing it twice. So it has to wait for a second before it sends a signal "this is a single press" to the OS. I don't believe it's caused by phone waking up from sleep because it happens every time and it's consistent.

If my guess is correct and I'm not using "Google Now" or "S Voice", it would be great to remove this delay. How do I do that? Is that even possible?

In my opinion, it is annoying. The moment I press the home button I'm expecting an instant response by showing up the home screen as immediately as possible.

EDIT:

I already disabled all the apps that could possibly make use of double pressing the home button like "S Voice". Now when I open an app to foreground, pressing the home twice doesn't take me to home screen but instead nothing happens. That means there's no longer any function that is currently bound to double pressing the home button. If I press the home button once, it still takes a delay before it takes me back to the home screen.
An important navigation mechanism in Android is called a long press. You perform a long press by pressing and holding down a physical key or touch (capacitive) button. If the long press is successful, the phone will provide some physical feedback such as vibration, and the long press action will take place.

You can use a long press to open menus that are specific to the active item - think of it like opening a context menu on a PC. For example, performing a long press on a song in the Music player application in Android 2.2 will open a menu that allows you to add or remove it from a playlist. To perform a long press that opens a context menu, press and hold the select key or the trackball.

Yes you can increase/ decrease  the time duration of the home button under the settings>Accessibility scroll down to Mobility.

Select your desired time gap from the three  -Long/short/medium