Help with 22 months toddler girl with community paediatrician referral.

HI there,

I am a bit lost regarding what to do with my daughter. She is 22 months old and shows almost no joint attention. Otherwise she is social, good eye contact, happy, affectionate, not many tantrums and sleeps the night through. She eats well and she likes to play peekaboo, sings a lot and really well. She waves and says bye, and she smiles and babbles all day. She is been raised trilingual and has few words for her age. She attends full time nursery, and they have told me they have not seen any red flags and that she is actually gaining vocabulary (in english), that she is very social, plays along children (no social play yet but this is normal) and she understands and follows instructions. 

However, the lack of joint attention together with delayed communication is driving me mad. She's only had two moments of shared attention with us, which happened last week, and in the past weeks has shown me some toys like 3times. She also does some pretend play.

I have done online M-Chat and depending how harsh I am she shows 50% risk. I have done other screening tests and show always low risk. I went to the GP last week and burst into tears because I am convinced she has ASD, probably mild. I felt ridiculous and terrible. He told me however he would try to refer her. I am based in Camden, so does anyone know how it works here? Should I seek a private assessment? They cost a lot money but could make big effort if that would actually help to clarify and easy the process. 

Thanks