I can understand why the child's birth mum is so hurt by this, and I do think that it was a bad idea to let the child start calling the OP mummy so quickly - before they had even moved in together, iirc), but I do think that MindReader makes a good point when she says that the child's wishes should be respected. And since the decision to allow it (whether too early or not) has already been taken, that ship has sailed, and now the OP, her dp and everyone else concerned has to deal with the situation as it is.
I do think the best idea would be a bit of research to find some alternative nicknames or versions of mummy, and ask the child to choose one - as a 'special name for you to use'. If I recall correctly, the child is old enough for an explanation as to why her birth mum is hurt by her calling someone else 'mummy', and to understand why a different, but equally special (to her) name is being suggested.
On a more general note, I do think that 'mummy' and 'daddy' are far more than just a biological term - conceiving and giving birth are just the start of all the things we do as parents, and someone who is there, a constant presence in the child's life, feeding them, comforting them, playing with them, disciplining them, teaching them, loving them - that is a parent/mummy/daddy, isn't it?
I realise IABVU for admitting this - but I sometimes watch Jeremy Kyle, and whenever I see him tearing a man to shreds for allowing a child who is not his, biologically, to call him daddy - when the biological father is not, nor ever has been involved in the child's life, even to the extent of providing financial support, and the other man is a constant presence, providing for the child (financially, emotionally, socially etc). In circumstances like that, I think the man who is there for the child and the child's mum is being far more of a 'daddy' than the biological father is - if that makes sense?