Hi Pumpkin.
There are two separate things to do. One is the medical route for the diagnosis, and the other is the educational route for the Statement.
After the law changes next year, it is supposed to all come together as one, but, for now, it's two separate pathways.
The school can ask the Educational Psychologist to come in and observe and work with your child, and, if the meeting after that agrees it's the right thing to do, can instigate what they call 'Statutory Assessment'.
or
As a Parent, you have the right to make a parental request for Statutory Assessment (you make the request to the Local Authority).
An Ed Psych will want to see what the school is already putting in place though, so, at you meeting, you need to ask the school what adjustments they are currently making to meet your child's needs, and can you have a copy of that IEP (Individual Education Plan). If they haven't got one, then you need to ask them how they are documenting the additional support and the impact it's having on your child.
If you find the school is not supporting your request, then it might carry more weight if you have a diagnosis. To get this, you have to be referred to a CDC (Child Development Centre). Again, the school can do this, but around here we find a quicker route in is to get your GP to refer you. The medical diagnosis needs to come from a Paediatrician, but does not, on it's own, provide you with automatic support in school, you need to apply for that separately (see above).
Feel free to come back and ask about anything that's not clear - there are loads of lovely MNers with this type of knowledge.