Something seems to have gone a bit odd here. The paed has done a report which he hasn't copied to you, Lisa, and then says that your DS is on the spectrum without actually waiting to hear what the SALT has to say. I also think that the use of 'developmental delay' suggests the dx process hasn't been finished as it is a bit of an umbrella term.
It sounds to me as though the dx process has got muddled with the statementing process and what the paed is doing is to help the statementing or at the request of the ed psych and not to get a formal dx. Just my guess of course.
What you do about it I don't know. I think talking to the paed might be the best start. Failing that talk to your HV about what the process of dx should be in your area - it may be that you have to start again, or at least get proper reports done and an official dx.
It is all very well saying your DS is on the spectrum (and not even copying you in!) but it isn't that simple and something could be missed if the full assessment hasn't been done. For example, it became increasingly apparent that my DS had dyspraxia as well which wasn't even mentioned when the dx process began and it turns out as he has got older, that is much more of an issue than the ASD. I'm not saying your DS is the same but it is worth doing the dx properly because the dx can be complex. Given that you haven't see the SALT I don't see how all the information can have been gathered.
Also what happened when you saw the paed? Did he or she do a full range of tests or just chat to you and observe your DS?
Also, are you sure that you need a statement? I only ask as the nursery are saying that they don't do anything for your DS but he may need extra pushing. That doesn't sound like the basis of a statement. You can get that from a teacher who is aware of your DS's condition. I think you need to ask what the statement is for and what they hope to achieve by it. For your sake as much as anything - it is a long process to go through for nothing and not all AS children need one. There might not even be any additional money in it for the school to get a statement - not all statements provide extra cash.