My friend Rob Dougall and I ran this for my Birthday last year. We had to go around some of the parks that were either not open or closed when we passed through but in all instances we took a longer rather than a shorter path. Here's a Strava track: https://www.strava.com/activities/1028291459 . Here it is on Garmin Connect: https://connect.garmin.com/modern/activity/1787023699 . Our duration for the whole run was 17:40 but 18 minutes was running to and from the trailhead - so let's go with 17:22.
There are photos on my Strava track. It was a fun way to spend the day and see some of London I would have never seen otherwise - but I enjoyed running around Mount Rainier much more in terms of scenery. It's a pretty urban slog. I think beating this FKT would be relatively easy for someone who is in good shape and has a nice summer day to spend circumnavigating the capital - our pace was not super fast. Summer is better as paths are dry and days are long - this trail can get super muddy in the wet weather.