The most comprehensive history of this tunnel can be found in the book "The Story of the Mersey Tunnel, officially named Queensway". This is a facsimile of the book published in 1934 to commemorate the opening and was re-printed in 1994. If there are any left, you can get one from the tunnel HQ at George's Dock. It used to cost £5.
The unfortunate consequence for Birkenhead, is that this began the isolation of the communities around Chester, Ivy, Pilgrim Streets and Waterloo Place, which continued until the death knell was sounded by the tunnel flyover scheme @1970.
Look at all the surrounding buildings and compare it to now....