I think it's very sad that the nursery key worker has written this report and hasn't shared her concerns with you before giving it to you. It would have been very helpful if she had discussed it with you beforehand. However, yes, if the key worker is describing your DD accurately, then you are right to be concerned and I would take that report to the GP and ask to be referred to a developmental paediatrician for a developmental assessment.
A developmental paediatrician can use something like the Griffiths scale to assess your DD. She wouldn't even know she was being assessed. She would just think that a nice grown up was showing her some toys with her mummy. But during that time, the Paed would be noting down how she responded to his/her attempts to get her to do certain activities. It's very simple things like building a tower of bricks, putting a teddy on a chair, putting some puzzle pieces in a puzzle board, pushing a car along the floor, etc., as well as whether they can keep her attention, whether they can get her to point at things, say certain things in response to a cue, etc. From that, they can work out what developmental stage she is at in every area of her development.
Most children are more ahead or behind in one or two areas than others. You know she's quite behind in speech, but they will be able to work out if she's also behind (or ahead) in her understanding of language. They will look at her physical skills, both gross motor (walking, jumping, etc) and fine motor (holding a pencil, etc). Then social interaction, which may be behind, if she isn't making eye contact.