In my case, as a British citizen married to a European living outside the UK, I had way fewer requests/demands for documentation.

All I needed to do was submit my own details, and a copy of both the marriage certificate and birth certificate for the child.

The assumption was made that as a married couple the child had me as the father, and as I'm British then so were they.

No references at all to grandparents, or even the mother's details, in the online wizard I filled out. Quite a quick process, which reminds me I probably need to renew their UK passport since it has expired a couple of years back.