When you turn off Wi-FI and mobile data app starts immediately, but once Internet connection is on server respond time kicks in and that is the thing I cannot speed up myself.
You said yourself it uses xiaomi.eu global rom, but we never built global rom for this device, so I highly recommend to install our dev/stable or even official as it is already unlocked, and yeah you would need TWRP.
Custom theme might cause it, all themes are designed for phones and you should not use custom lockscreen. Just go to Themes > Your profile > Custom > Lockscreen > Default.
Perhaps android base cause the issue here, however it should not. You are on oreo I guess and latest is Pie. I cannot remember all fixes since November so I would suggest to download manually from our download to save your time.