Use the Flight Connection Centre at Terminal 1 to have your hand baggage screened and take a bus to T3. You can get your BP from TG Transit Counter at T3 (I don't remember who is handling TG, it used to be SK but now it could be Servisair). At T1 hand baggage screening facility, there might be a guy asking what your final destination is (as UK/Ireland connecting passengers must clear Immigration control at LHR), but she/he is not checking BP'es (just in case - have a copy of your e-ticket handy).
It is really a straight forward process and in the evening LHR is a good airport to change aircraft.