A long, tortuous issue, where no progress has been made, despite large amounts of effort. Godalming High Street was recommended for adopting as a trial two-way working for cyclists in 2001, thanks to Alec McCalden's excellent presentation to the Planning Inspector at the time.
However, the trial was never carried out. Cyclists continue to use the street in both directions, as there are plenty of ways to get out of the way of oncoming cars.
The City of London has introduced dozens of contra-flow cycling schemes on narrow streets and lanes across the City. A third of all the cyclists are now using the contraflow. There has been 1 cycle crash three years post-introduction (there were 2 in the three years before). See here for more details: https://www.ucl.ac.uk/transport-institute/pdfs/Cheung_slides.pdf