I just got back yesterday from London/Dublin and was so confused as to how I could have missed passport control landing at LHR from DUB. I was very relieved when someone finally had an answer for me that I hadn't missed it, it wasn't required.
When I went through in Dublin, the signs had so much writing on them that I was overthinking it and asked an employee for help. She was rude "There are signs, read them", but then laid it out clear as day, non-EU people go through the green gate. I hope her day got better as it went along!
I've always been really lucky collecting luggage at LHR and have been able to get bags pretty quickly - not the case for everyone though so I always plan for more than I have waited in the past. I take the tube to my hotel so have never really needed a firm itinerary for this portion of the travel.
Flying back from Dublin to the US, the check in agent will give you a time to make sure you're at the US pre-clearance facility. Personally, it was an overly generous window for my flight this week as there weren't a lot of travelers in the airport. My flight was 12:50 and they said to be there by 11:05. The Aer Lingus lounge staff said 11:30 would be enough time - they were right in my case.
Have a wonderful trip!