Yes, only when arriving on a domestic flight (UK only, so doesn't include JER or DUB) would you not have to do security again on arrival to T5, and only when you are connecting on to a flight departing from T5. So even if you were connecting to a T5 flight you would have to do security when coming from JER anyway - basically JER is the same as DUB for the purposes of this guide.

Even if you were arriving on a UK flight, once you do a connection to any other terminal you would mix with international arrivals and have to do security at the onward terminal.

For your connection you will arrive around A23/A22. You have the option of going right and heading to the airside connection buses, or heading landside and taking the HEX/crossrail to T3. Personally I would probably do the latter in your case unless your connection is very tight.