If you say George's mum should I use a capital for mum?

Answer 1

No, the noun 'mum' is a common noun, a general term for any mother of anyone.

The noun 'mum' is a proper noun only when addressing a specific person, such as your own mother, in which case it is the name you call your mother.

Answer 2

Yes, you should use a capital letter when referring to someone's mother as a title, such as George's Mum. It is a sign of respect and correct grammar to capitalize titles when they are used as proper nouns.