For European flights at T5 check in will close -45 minutes before the departure time. You need to go through security -35 minutes before or you will be bounced back. I would aim to be there 120-90 minutes before the flight. T5 is very large and it's a long walk anywhere.
If you have status with any One World airline you may be able to use the business class check in.
It might also be worth thinking about buying a business class/Club Europe ticket which allows you dedicated check in, fast track security and 2 bag per person baggage allowance. You also then get lounge access.