It depends where you live. I went to the Brindabella Hearing Centre here in Canberra and they deal with diagnosing all types of hearing/processing/speech issues. Your GP might know of something similar in your area, but they wouldn't have the equipment to do it themselves.

To give you comfort, it is a very treatable issue and often just requires understanding and slight changes in how you as a parent phrase things etc. And catching it early will avoid many of the issues they may have with schooling etc.