A look at the LHR website would have revealed how to do the connection:

Link: <http://www.heathrowairport.com/portal/controller/dispatcher.jsp?CiID=10b448a504093110VgnVCM10000036 821c0a____&ChID=217fbf54776d4110VgnVCM10000036821c 0a____&Ct=B2C_CT_GENERAL&CtID=448c6a4c7f1b0010VgnV CM200000357e120a____&Ch=Flight+Connections&ChPath= Heathrow%5EGbl%5EHeathrow%5EGeneral%5EAirport+info rmation%5EFlight+Connections&ChIDPath=9a485109350d 3110VgnVCM10000036821c0a____^d7b3b727436d4110VgnVC M10000036821c0a____^e734b727436d4110VgnVCM10000036 821c0a____^7544b727436d4110VgnVCM10000036821c0a___ _^8454b727436d4110VgnVCM10000036821c0a____^217fbf5 4776d4110VgnVCM10000036821c0a____>

I would not suggest the landside connection even though SEs in economy are supposed to get the pink Fast Track envelop. However, you can generally get through customs in 15-20 minutes, but add another 10-minute to get there from the gate. Bags will go right through so no need to wait for them. There is no terminal shuttle bus on this side of security so you have to take the tunnel, which adds another 15-20 minutes. Security lines at T1 are variable, so anticipate 20-minutes. So that pretty much eliminates the Arrivals Lounge. But if you can cut down some of this time you can use the STAR lounge which has showers.

Since the UK is not part of the EC immigration/customs zone you should use the airside transit and not pass through UK immigration/customs, take a shuttle bus between terminals, and get right to the airside gates at T1.