Battle-eye understanding.

When battle-eye detects a cheat, does it immediately ban you? OR is it like VAC (where it has a two week delay after the cheat has been detected)?

Battleye claims to do banwaves/ delayed, so yes. I wouldn't be sure though.

I've been instantly banned before (within 10 minutes of joining a battleeye server) and i've also been banned after days or weeks. I think it depends really.