Searching for my Grandfather's family

Hello Everyone, my name is Michaela. I'm searching for my mother's real father's family. I don’t really know much, as my mum didn’t find out her dad wasn’t her father until just before he passed away. My Grandmother had passed some years before him, all I know is that my Nan was with a Greek sailor before my granddad (the man who brought my mum up). My mum was born in July 1945. She told me of a shop where the owners would always say "here's our littlest princess" and gave her ice cream. I think it was in the town of Greenock, Scotland. I’m guessing that was just after the war until roughly 1955. When my family moved to Bath Somerset, unfortunately mum has had a severe stroke and can no longer communicate with us, so I can’t ask her anything else. So any information anyone could give me would be amazing. Sorry I can’t give anymore info.