Christ the King in St Anne street (parallel with Beckwith Street) would be the protestant one. I thought the catholic one would be "Our Lady's" as that is (was?) in the same area. You could always contact below as they are the main one for the Birkenhead area and might be able to help.
St Werburgh's R C Church
Denomination Roman Catholic
Grange Precinct,,
Birkenhead
L41 2XZ
Tel: 01516479124