How close are you to the character in third person? If it's a bit more distant and neutral, you can call her Mother in narration and Mummy/Mommy in dialogue. It's really your call, I think both can work.
In the first example Mummy should be capitalized. In the second example it could actually go either way, so capitalize it if you want.
It's quite distant POV, I never write his thoughts per se. Ok thank you, I wasn't sure whether the title must be the same in speech and in narration. I'll be more careful about capitalization.
