Hi refeel,
Welcome to Apple Support Communities. To help with this delay in the screen mirroring, we suggest resetting the NVRAM on both computers if applicable. Doing so can help with certain display issues. Follow the steps here to do so: Reset NVRAM on your Mac
How to reset NVRAM
1.Shut down your Mac.
2.Turn on your Mac, then immediately press and hold these four keys together: Option, Command, P, R.
3.Release the keys after about 20 seconds, during which your Mac might appear to restart. For example, you might hear a startup sound more than once, or see the Apple logo more than once.
4.When your Mac finishes starting up, you might need to adjust any system settings that were reset.
Let us know the outcome, and we can go from there if needed.
Have a great day!