Technically, the footpath is strictly for pedestrians, you're not allowed to cycle on it, I know, I was shocked to find that the authorities would rather you mix with the fast traffic on the bridge! (unless the rules have changed?)
I think (not 100%) that the sign on the footpath is the blue and white advisory "CYCLISTS DISMOUNT" rather than something you
must do. Cycling across it is the norm anyway from what I see.
After that, I guess the best route would be through the town centre and into Astmoor and Manor Park and up through Daresbury, maybe via Sandymoor (Keckwick Lane)? Someone more experienced with Runcorn might be able to help.
Astmoor and Manor Park will be fairly quiet from my experience but the roundabout on the Daresbury expressyway (Evenwood Farm) is a bit hair raising, as is the expressway itself.
EDIT: Just to add; to get to Astmoor I'd come off at the first exit and keep left towards the town centre then take a right onto High St to meet back up with the A533 at Victoria Rd. You can then get on and straight back down the slip road (I.e. keep left) of the expressway for Astmoor Road. That would cut a busy chunk out.