I've got 1:45 between arriving at Heathrow T5 (Edinburgh to London) and a connecting flight at T3 (London to Philadelphia).
I'm nervous that I have to get to T3 and go thru security again in that amount of time. Are the free shuttle buses reliable?
Is using the Tube better?
All you need to know:
http://www.heathrowairport.com/heathrow-airport-guide/travel-between-terminals
You are within the MCT for this transfer which is 90 minutes. So should you miss your second flight you should be accommodated on a later one, presuming you are on a single ticket.
You will need to go through security again though as there is co-mingling of travellers taking the connecting route (if you had been flying out again from Terminal 5 this would not have been necessary). However, you connect airside if you follow the route in the link given above.
Don't forget to have all your electronics charged up and ready to power up. In fact, having them already powered on would probably be best...
Is using the Tube better?
No.
You want to stay airside. Just follow the signs for flight connections.