I completed this as a circular route heading clockwise, starting and finishing on the golden plaque (see pictures here & on the main Route page) in front of Buckingham Palace, and following the square blocks set into the pavement which mark the route frequently and at every turn/road crossing.
I'll note that to make this a circular route I did not use leg 10 per the TfL website. I also note that the route I followed does not 100% match that listed online, for two reasons:
1 - Closures. With so much building work going on in London, several short sections of the route were closed. In these cases I followed signed diversions, which were either as long or longer than the original route. It is unlikely on this route that 100% of it will be open at the same time, so I expect future attempts to also deviate very slightly as well.
2 - In some cases the online route and that marked on the ground by square Jubilee Greenway paving slabs does not match. In these cases I followed the square slabs on the ground.
Other comments on my trip (by mile):
9.0 - Brief stop to pick up McDonalds breakfast
13.5 - Closure near Pudding Mill Station due to building works, doubled back on myself to find signed diversion
20.8 - Woolwich foot tunnel, at both ends I used the stairs rather than elevator
22.0 - Blocks on the ground took a different route than that on online map, which has you cut down Ruston road
26.6 - Footpath alongside the Thames closed here, had to take a U-shaped diversion round and back to the river
27.4 - Went a few hundred yards perpendicularly off route to pick up more water and food at a Satrbucks by Cutty Sark, rejoined at same point I left
30.5 - Missed a turn here, had to double back
37.0 - Thought I had turned too early here, doubled back and found I hadn't
37.3 - Parliament Square barriered-off by Police due to the Stop Brexit march taking place on the day, took the next side street to rejoin the route at corner of HM Treasury
Thanks and let me know if you've any questions!