As promised (https://community.ricksteves.com/travel-forum/england/six-days-in-london-please-critique-my-itinerary), here is my revised itinerary. I hope it looks better.
I'm staying at the Premier Inn County Hall, close to Waterloo Station and Westminster Bridge.
As before, anything in bold is set in stone. Everything else can be moved or canceled. And anything in italics is a high priority for me.
Sunday 22 June
- 4pm: Arrive LHR, public transit to hotel
- Evening: Go for a walk in St. James Park
Monday 23 June
- 10am-noon London Walks Changing of the Guard Tour
- Afternoon: National Gallery and National Portrait Gallery
Tuesday 24 June
- 8:30 am St. Paul's Cathedral (entry timeslot until 10am)
- Rick Steves' Historic London Walking Tour (starts at St. Clement Danes Church; ends at London Bridge; estimated time 3 hours)
- 7:30 pm Romeo and Juliet at Shakespeare's Globe
Wednesday 25 June
- 9:30 Westminster Abbey
- Rick Steves' Westminster Walk (starts at Westminster Bridge; ends at Trafalgar Square, estimated time 1 hour)
- Saint Martin in the Fields Church (there a 2pm guided tour on Wednesdays)
- 7:30 pm Oliver! at Gielgud Theatre
Thursday 26 June
- 9am Tower of London
- Afternoon: British Museum and British Library
- 7:30 pm Back to the Future at Adelphi Theatre
Friday 27 June
- Thames River cruise to Greenwich
- V&A (open til 10 pm)
Saturday 28 June
- 9am Houses of Parliament tour
- Tate Modern
- 7:30 pm Academy of Saint-Martin-in-the-Fields Concert in Basingstoke
I did a little more research on visiting the Houses of Parliament. It seems to be correct that you can visit either House for free when they are in session and watch the speeches/debates, but I couldn't find anything about wandering around on your own. They don't offer guided tours at this time of year, but they do have a multimedia/audio tour with timed entry (and it's not free), so I booked that. I have to wonder, though, if people who enter for free from those who are doing the audio tour. Maybe they check to make sure anyone wandering about is carrying the multimedia device?
Sunday 29 June
- 11:45 am departing flight out of LHR
Thanks again to all who offered suggestions on the first iteration and for any additional feedback you can offer.