Both school and GP can refer, however you may find it easier to approach the school first as they will play a significant part in the assessment.
In my case (potential Aspergers) I asked for an appointment with the Senco and teacher briefly explaining why I had concerns. At the meeting I took along my concerns, what I was doing at home to assist and strategies I'd tried so far. This resulted in referrals to speech & language, occ health, paediatrician and educational psychologist.
It takes time to get all these appointments, so be prepared for that. DS was diagnosed with ASD in just under 12 months.
He was assessed for ADHD when we lived in the Middle East last year and he was borderline (impulsive/hyperactive). I think his behaviour is getting worse and he is struggling to form friendships,. His school work is getting worse and he screams and cries when he is asked to do homework. He can not sit and concentrate on anything for any length of time and bounces from task to task like a young puppy.He talks constantly and asks the same questions time and time again. He gets upset when told he can't do something that in his mind he has already decided he is going to do. I'm worried about him as his self esteem is at an all time low. He told a teacher at school recently that he wanted to throw himself down the stairs and break his leg as he didn't want to go to school anymore. My husband isn't very sympathetic and thinks he's just a difficult child so shouts at him constantly but I truly believe there is more to this.
School SENCOs are very variable in their knowledge and responses. There are quite a lot who dismiss parents' concerns for very poor reasons and schools are sometime unwilling to bring in Ed Psychs, etc because they have the pay for these services now.
Your best bet is to approach your GP and ask for a referral to a community paediatrician who can initiate a neurodevelopmental assessment.