There is a (much better this year, and free) app available which I have on my phone.
I haven't looked yet but usually the Foreign and Commonwealth Office (just down from Downing Street) is open at these events and I haven't experienced particular crowds there. It is a very impressive building and the visit to the room where the Lateran Treaty was partly knocked out is exceptional.
See if you are near Blackfriars station any time during the weekend. A visit to Apothecaries Hall is well worth working out the fairly restricted hours it is open during Open House.
The Gherkin is one I think I shall never see the inside of - slow queues all the way around the block and then some every time I have tried - but not far from there is Lloyd's of London with its inside out architecture and Lutine Bell. Another good and easy one.
Usually plenty of churches are open, some with bell ringing tours, and bags of modern architecture as well as many older.
The problem is that there is months of stuff to see and only 2 days and 2 legs to do it. Pace yourself and try not to run all over town (says he who never does).
Oh, and if you get to Biggin Hill in Surrey there are usually a couple of Spitfires about. Maybe visit a WWII Home Guard defensive emplacement to guard the field.....
Anyway - have fun.