Depends what time you need to get there and exactly where in Birchwood. Walking between Bank Quay and Central is pretty easy if that is the route it is sending you. Alternatively it is about 30 minutes on the bus, from the interchange, but that is next to Central Station anyway.
As far as I'm aware no operator in the North bans bikes at peak times but there is a limit to how many they can carry so you may not be allowed to take your bike on board if all the designated spaces are occupied. Although when you are boarding at the origin station the easy solution to that problem is to arrive early.
Folding bicycles, e.g. Bromptons, are permitted at all times on all TOCs. Or the OP could buy a second bike for cheap and leave it at Warrington all the time. A lot of London commuters do that.