Have flown DFW-LHR-SNN and back every couple of months for several years. I'm BA silver which equates to OW sapphire and always have had access to the EI T2 lounge whether arriving to LHR on AA or BA. Also, always had access to SNN lounge on return trip. All travel either WT+ or PE and booked with BA. Never have bothered to go landside at LHR as I keep my connections short.
As an aside, if you have US Global Entry then you will want to get your westbound T/A boarding pass reissued at LHR as the EI SNN issued boarding passes will not indicate your GE status.