One thing I've learnt on my SEN journey (my DS is now 13) is that you need to be really proactive: don't rely on what anyone else tells you or says they're going to do. Also, always follow every conversation up with an email so there's a paper trail.
Has your DS's nursery said why they've not applied for an EHCP? As others have said, you should just apply for one today. The whole process should, legally, take 20 weeks but it could take much longer. If you need to appeal, it could be up to a year.
All the information you need is here. I did it myself. It can feel daunting but there are people who can help you, both on these boards and charities such as Ipsea and SOS!SEN.
https://www.ipsea.org.uk/asking-for-an-ehc-needs-assessment
Have you identified any specialist schools you think would be suitable for your DS? If not, I would get looking now. Then call them to find out what their entry criteria is. Some may take DC if they're working towards an EHCP/diagnosis - some won't.
If you have an EHCP, you don't have to just look at schools in your borough. Again, info here:
https://www.ipsea.org.uk/pages/category/choosing-a-school-or-college
I didn't think my DS would ever be able to go to school but with the right support, he is doing pretty well.