Do states in the US not have a process for civil annulment?
RC annulments by themselves don’t count for anything in English law, but then English law doesn’t recognise per se RC marriages in the first place. However, you can certainly get a marriage annulled in the civil courts. Presumably you’d then have to request the [Roman Catholic] Church for an annulment as well…
Incidentally, I don’t believe that the Church of England has any power to annul marriages in England, despite it being what one might call the “state religion”, and despite its clerics pretty much having an inherent right to perform legally-recognised marriages. (Henry VIII’s marriage to Anne of Cleves was apparently dissolved by Act of Parliament; I dunno about the others.)