On the specific question about being sent to the lounge after check-in, I don't understand why this would not be possible. I flew QR this week from LHR T4 and checked twice with QR people (at the check-in and at the lounge) on whether the wheelchair would be sent to the lounge and both people confirmed yes.

Therefore, Singapore Airlines LHR should be asking the LHR wheelchair assistance personnel to have you sent direct to the SKL rather than the holding area for Y pax. I have no upcoming travel ex LHR on SQ until August so cannot check until then for you with SQ personnel directly. Maybe someone else on the forum can assist ?