On the way back from the US to Dublin, I knew that there were no showers in the BA UK/Ireland Lounge and so immigrated at Terminal 3, and went to the Arrivals Lounge.
There was no line at the (still) operating IRIS machine at Immigration despite several large aircraft arriving at once. There was a line for UK, the ePassport Gates and the Non-EU passengers.
The Arrivals Lounge is on the second floor, reached by turning right after you leave the Duty Free shop after Customs, catching a lift to the second floor.
On arrival I was met with a helpful member of staff who arranged for me to have a shower room. A collegue showed me where to go, and then they checked that I knew how the room operated.
In the room is a large multi-level shower, basin and toilet.
After my shower I popped out for a sit, and some breakfast. There are about 40 seats and there was plenty of room despite the lounge being shared with Qantas and Cathay passengers.
There was hot and cold food, a good range of juices, coffee, tea and water.
You can buy your way in to the lounge for US$90.