I'm afraid it doesn't. Having an EHCP makes a child excepted. Simply having SEN does not. The information is on the legislation site at The School Admissions (Infant Class Sizes) (England) Regulations 2012. The paragraphs referring to SEN are 2 (pupils with an EHCP - referred to as a statement here as they haven't got round to updating this on the legislation website), 10 (pupils who are registered at a special school but attend an ordinary primary school for some of their education) and 11 (pupils who are educated in an SEN unit forming part of the school who spend a minority of their time in the infants class). On the information you have given, your child does not fall into any of these categories. He will be an excepted child if he is admitted via appeal, but that doesn't help you.
If this is an infant class size case, you will only win if a mistake has been made and your child should have been admitted. In that situation, you should win, and the other appeal is only relevant there was also a mistake in that case and the appeal panel concludes that the school can't possibly accommodate both children.
If it is not an infant class size case, you will have a better chance of success. However, SEN will only help your appeal if you can show it means he needs this school, e.g. they have special provision for his needs that is not available at the allocated school. An argument that he needs to attend this school because he has attended the nursery here will only fly if you have evidence from a medical professional to support it.
Childcare arrangements are irrelevant for appeals. Appeal panels are not allowed to take such arguments into account.
If it is not an infant class size case, the other appeal may impact your chances of success. If the appeal panel decides that both of you have good cases but the school can only accommodate one of you, they will compare your cases to decide which to admit.