text: This one was interesting. It turns out that "Old Bells Line of Road" was not part of Bells Line of Road (which ran west from Clarence Station down to Bells Road where the water treatment plant is). This road mostly follows what is called "Maddocks Line of Road" on the 1890 parish plan. At some point the "Old Bells Line of Road" has been applied to this road and it is what the Council seems to call it. 
  
  "Newnes Forest Road" gets used much less online and the only real evidence seems to be the sign you photographed. I found an older photo that showed the second board on the sign used to say "Clarence Sawmill" which would suggest who put the sign up in the first place. 
  
  Don't know what you want to do about this one but based on a recent picture (https://goo.gl/maps/3K91rnqoAkZKhUp97) waiting a couple more years may solve the problem as there will be no more sign.