I investigated this and it is very, very confusing. Here are some certainties: all signs in Warrington strictly forbid cycling. Signs between Walton and Daresbury are mixed up and I expect this to be an error in signage rather than being deliberate. There are signs which ask cyclist to give way and signs which strictly forbid cycling.
The website explicitly says that cycling is only permitted on certain stretches of the road and none of them covers Warrington or Runcorn-Warrington path. There's some stretches in Warrington and Preston.