Ok, for now here is the deal,
My GS certificate (both course completion and exam results papers) are granted by CATS. As they are authorized to perform PPL exams on site, not likely as ATPL exams, they signed and stamped my documents.
When my school here applied to Italian CAA to ask for a skill test, the answer was "We need to confirm these papers, because they are not UK CAA approved" and my school was not optimistic by telling me this.
But the thing is, these papers shouldn't be approved by any CAA. All ground schools are authorized to perform PPL exams.
So I've asked them if we have any chance to get my papers approved by UK CAA somehow. Not because of some law/regulation says, but to get around with the never ending demands of Italian CAA.