I think I'll close this now, as it's getting rather complicated!
Liverpool, Newcastle and Birmingham have both a Cof E and a RC cathedral.
London has 4:- St Paul's [CofE];
Westminster [RC]; [NB CofE Westminster Abbey is technically not a cathedral];
Southwark [Cof E & RC]
Derby, Manchester, Leeds, Bristol and Nottingham have only one - [Derby, Manchester and Bristol are CofE; Nottingham and Leeds are RC{the current Cof E equivalents are Southwell Minster for Nottingham and Ripon for Leeds - although that may change}] and, technically, the RC cathedral serving Manchester & the NW is in Salford.
Coventry I'm counting as one - the "new" St Michael's of the 1960s replaced the adjacent ruins.
I know of three others with 2 [one CofE, one RC]:-
Sheffield, Norwich and Portsmouth
So, unless anyone out there knows differently, I'm going to declare Clagmonster the winner with 4. Well done!