I enquired a few months to the tunnel police about riding through the birkenhead tunnel (queensway)and they started reeling off all times you can do it at certain times of the year (different for suimmer and winter I think). There were alot of dates and times thrown at me so I've kind of forgot now but it can definitely be done. You'll have to make a phone call to double check.
Driving through the tunnel the other evening at about 8pm and there was a cyclist in there.
Queensway more central but not by much.
You should have no problem at 4am but double check first incase new prohibitions in force.
Add you can take bikes on the 'underground' (not referred to as the 'tube' up here!)with all trains having cycle storage albeit a bit minimal. I think that they start at around 6am so this will probably be no good to you anyway.
Big descent, equally as big climb!