I see that both “growing slowly” and “growing up slowly” are accepted. In English, these have different connotations: “grow” = physical development, “grow up” = behavioural/mental/emotional development. Does the Czech expression cover both of those?
It does. Although there can be some subtleties and other verbs like "vyrůstat" and "dospívat" involved.
I wasn't allowed "Your daughter is slowly growing up" -- shouldn't that be a correct translation too, or am I missing something?
No, that is not correct.
"Your daughter is slowly growing up" means that your daughter is slowly becoming a woman. I would translate that as "Tvoje dcera pomalu dospívá."