A study by consultants last year found that electrifying the railway from Wrexham to Bidston , near Birkenhead , would bring major benefits because the electric trains would be allowed into Liverpool’s underground stations (link below.)
http://icwales.icnetwork.co.uk/0100news/tm_headline=wag-to-be-urged-to-rethink-railway-link-delay%26method=full%26objectid=19904996%26siteid=50082-name_page.html

Diesel trains are banned from the tunnels under the River Mersey and the city because of lack of ventilation and fire risk.

Passengers from Wrexham , Buckley , Hawarden and Shotton are forced to change trains at Bidston , so many opt instead to drive to a park-and-ride station on the Chester-Liverpool line.

Their cars add to the heavy traffic on the A494 , which the WAG plans to widen to seven lanes north of Ewloe. But it has put the railway electrification back to the medium term , 2014-18 , or the long term , from 2019.

The move has surprised Merseytravel , the WAG’s partner in the project , which had pencilled in the project for 2011.

Proposals like this tend to be looked at more favourably if you have a public body such as Merseytravel on your side (link below.)
http://www.merseytravel.gov.uk/who-we-are.html

As the original post states , there would be major benefits to the area if the Wrexham to Bidston service was extended into Liverpool.