I saw that my search console is not giving any results although I verified the website (www.caglio.co.uk).
any idea how to solve this?
You have no google indexed links for www subdomain, all of your google links are to the main domain so the info shown there seems correct. To check search site:caglio.co.uk and then site:www.caglio.co.uk
that's what I see when I search it as you said...
Yes, that shows you have no www.caglio.co.uk links in google, only links to caglio.co.uk so the info in search console is correct, verify the non www version instead.