You don't need a diagnosis to get at statement, you just need to prove your ds's needs and if the nursery are saying he needs full time 1:1, he will definitely need a statement.
Attila's advice is excellent, as always. Unfortunately you won't get a statement in place before September now, but I would highly recommend getting the ball rolling if the school is dragging their feet. Personally, I found it much easier to apply myself, as it meant all the communication came through me, so I always knew what was happening. I know it can seem a bit daunting, but it's actually a very simple process, if I can do it, anyone can. 
Has your ds only seen the EP? If so, I would advise going to the GP and asking for a referral for assessment. Again, as Attila said, EPs cannot diagnose on their own, it's need to be a Paediatrician or multi-disciplinary assessment team. In our case, ds was referred to the Community Paediatrician in the first instance and then referred on to the multi-disciplinary team.
As he is unlikely to get his statement through in time for a September start, I would recommend arranging an appointment with the school's SENCO to discuss his needs and make sure he has appropriate support in place in the interim.