If you fly domestic into T5 on BA and then out of T5 on international you dont need to go through security again at LHR assuming your bags are checked through to your destination and you have all boarding passes.
If you fly domestic into T5 on BA then out of T2, T3, T4 with a partner airline and some others on international and your bags are checked through to the final destination and you have all boarding passes, you have to get the airside transfer bus to the relevant terminal (you dont need a transfer ticket just a valid boarding pass also dont need to collect bags) where you will have to go through security again before entering the departure lounge in that terminal.
If you fly into LHR terminal ,2,3,4 or 5 on an international and your checked to your final destination ( baggage and boarding pass) and you depart on a domestic out of T5 you will use the airside bus to get to your departure terminal. and you will have to clear security at the terminal you depart from. If your flying in to T5 on international and out on domestic you still need to clear security at T5. ( dont need the transfer bus though.)
Hope that makes sense.
So for Eastern from Teeside there will have to be some through check in facility in place otherwise you will have to collect bags if you have any at the arrival terminal. Transfer with your bags landside. Check in again for your next flight and go through security. Hopefully something will be in place to avoid this.