I met my boyfriend's child (3 y/o) after 7 months together.
We don't live with each other - as one poster has said, I guess I am very much 'girlfriend' and not 'partner' - but he spends so much time with his child (which is fantastic for both of them), and fatherhood is such a huge part of his life, that after 7 months it felt natural that I should start becoming more involved with that side of his life if things between us were going to progress in the direction we both want them to. His ex had no issue with me meeting his DD so he slowly started introducing me.
The way we do it is on the weekends he has her, I meet them somewhere neutral like a museum or park and spend the day with them. Then they go home together and I go back to my place.
We are careful not to be affectionate in front of her. We didn't want it to be all like 'This is dad's new girlfriend, get used to it!' and then be all over each other. Understandably that would freak her out, so as far as she knows I am just one of daddy's friends.
Sometimes she tells us to hold hands as the three of us are walking down the road, so that we're all in a chain. That is super cute and makes me feel unbelievably happy (:o), but we only do that if she tells is to. Otherwise we are pretty hands off with one another.
There is no rush and I don't want to force my relationship with her father onto her before she's ready, so I am happy to continue being 'the friend they see on Saturdays' for a while.
Just because I am in a relationship with her dad, I have no automatic right to come into her home and start spending the night, being involved in her life, etc, without her fully understanding what the situation is and being 100% comfortable with it.