Definitely too risky ... T2 is pretty busy at that hour with lots of USA departures etc. and if flying EI you won't have access to any fast track security / priority checkin or anything to make it faster ( I don't think ?)
With just hand luggage and your boarding pass in hand from online checkin it might be easy enough ( but I find that if you're booked on codeshares on EI flights you often have to go to to the desk to get your boarding card )
Change onto the BA flight & travel with HBO or come over the night before would be more prudent.