You can easily get from South Mancs (sale/ chorlton / didsbury etc) using the TPT to Urmston, then Flixton, Irlam Locks, cycle lane next to A57, take Dam Lane in Glazebrook across the Moss in to the megalopolis that is Birchwood. That’s my commute, usually done twice a week, originally on a SS genesis io, nowadays on a Kona Sutra borrowed from Plumber ( 5 mins quicker on cx bike).
You didn’t say whereabouts in Warrington you’re heading too, but you can alter this route to work for you.
Be careful when tackling roundabouts in Warrington in the mornings – most of the locals have had that part of their brain removed that governs their ability to give way to bikes approaching from the right.