This is a very good question. Ultimately it's up to you but here is some advice.
Our scheduled trip is somewhat on the long side but if you can spare an extra day my advice is to arrive a day early. It's a long flight from Seattle and depending on what flight you get you will probably stop in Heathrow and then hop up to Edinburgh. You will probably arrive mid to late morning and be dead tired by the time you get to the hotel and checked in. No activities are scheduled until the early evening. Colm will have a group meeting where he will introduce himself, go over how the tour will be conducted and what expectations will be. He will also hand out our "Whispers" which will be covered in a later post. After the meeting we will have dinner and then collapse in bed.
The next day is designed to be relatively low key because you will still be pretty tired. On our trip to Ireland about half of our group arrive one or two days early so they were ready to hit the ground running when the official tour started. While those of us who stuck with the schedule were dragging the first day and a half.
Bottom line is go a day early if you can manage it.
By the end of the tour you will be a bit weary as well but London is a very interesting place and our schedule only allows for a taste of it. You could easily spend a whole week exploring the city so if you can stay then do it. Getting around is pretty easy once you lean the Tube system which takes about half a day.
GoAhead will work with you on air, lodging, and airport transportation arrangement should you wish to stay or arrive early.