3 year old beautiful boy

Hi,

this is all new to us as a family so hope I make sense, our 3 year old beautiful boy has some struggle, we are under speech and language therapy a consultant and his goes to a school nursery, we had a call today from the consultant who has referred him for a autism assessment, we should hear in the next few weeks weather or not they accept or reject him for a assessment, how long roughly does it take for them to be in touch? She has discharged him from her care now, if he gets rejected what happens then? Also how much is a private assessment we feel like if he gets rejected this is something we will look into to get him the help, sorry for all the questions i just have no clue who to talk with to try and get answers ( feel like nursery are just not bothered) thank you in advance for anyone who can help 

 thanks