I suspect it'll be a bit lighter than the liverpool>manchester train, as that was rammed to the point of having to miss a couple of trains in a row due to overcrowding at rush hour, and thats without a bike! But if you get the 10am trains or can get as low as the pre-8am trains you're generally safe for getting a bike on, but expect to stand with it and guard it 🙂
Also be prepared to be infuriated by the people on the train, the people near the trains, the staff who clearly can't be arsed collecting tickets and if you offer to pay by card expect to be shot on sight (apparently thats wholely unacceptable despite them having the machines). Oh, and you may need that axe if you're coming home after 8ishpm. Lost track of the number of drunks fighting I've seen on trains!