Looks like LHR T5 (and MAD for that matter) have been removed from the list of partner lounges that Admirals Club members (including Citi Executive primary cardholders, but excluding authorized users) are permitted to access. They definitely used to be on the list.

Having cleared T5 security, yes, you would need to re-clear security in T3 and again upon your return to T5 following the lounge visit. Additionally, if you left the T5 departures area by staying on the train at T5C, you would also need to have a BA employee reset your "ready to fly" status prior to being allowed up the escalators to re-clear T5 security, which from what I've heard takes some time and adds an additional complication.