This is going to sound super silly but I have a coach that gets into terminal 4 at 3am. I'm checking in online and then my bag drop at least 3-1 hours before the flight which is at 8.40am. Can I stay in the terminal until then even though I won't be able to get past security or will they kick me out until closer to the time?
I've never needed to arrive at an airport in the middle of the night before
Airside is everything AFTER security. You cannot get through security without boarding passes, which you cannot obtain until your check-in desk/bag drop counter opens.
So I think you can stay landside for a couple of hours, but be warned, it won't be pretty. Most airports have most shops and cafes shut overnight (especially Heathrow, as no planes land or take off between 11pm and 5am anyway), but there will be a few 24hr food outlets like Starbucks or whatever - maybe even with free WIFI.