ahh it really varies from area to area.
generally they really like schools to refer them. I presume that this is because the school will hopefully have picked up on any issues and will be able to get the SENCO to produce a report to go along with referral,
In our case the school was very reluctant to do anything, and so I went to the GP and asked for a referral (took 3 visits and 2 different GP's but we did get it done!). they did want to ask the school to refer first and contacted them, school said no, so we went back and GP agreed to refer.
So, I would say that it might be worth talking to school first. If they agree with you and are happy to refer then great.
If not then see your GP, explain school not willing to refer and insist on a referral via them.
You will normally get referred to the community paediatricians, although some will refer to CAHMS (depends what the issues are)
We were seen and assessed and given a diagnosis within a year. but it varies a lot and I know some people on here who've had a real battle with it.