As usual with LHR - you can never be sure. One day it works best with one route and then it doesn't.
As LHR is my local base, I tend not to have to use the connections and the one time that I did have to - I chose to go the 'manual' route. The manual route will take you down to the tunnels but they can be pretty fast, especially if you only have a hand case.
Basically, there is a hub that serves the tube station for 1-2-3 and each terminal has a tunnel leading to it. So the cross-over point at the entrance to the tube station can get busy but just follow the signs and the next tunnel will take you straight over to 3.
From emerging landside at T1, to entering the departures hall at T3, should take about 10/12 minutes but always time of day dependent.