a. Bishop C
Presumably The Bishops CoE and RC Primary School (North Springfield)
b. Springfield primary
c. Tyrlls
Tyrrells Primary
Which year group is your DS in? If you make a formal application for a school and are turned down, you automatically have the right to appeal, but if you are applying for yrs R - 2 and the refusal was based on Infant Class Size legislation, your chance of success would be close to zero. I would never say 'don't appeal', as appeal panels do sometimes uncover an error in the process which means that a place should have been offered, but this only occurs very rarely.
For a mid-year application, it may well be a question of which school has a place to offer, rather than you picking the one you want and your DS gets a place there. It would be sensible to contact the schools you are interested in, and the ECC admissions team to see if you can get an idea of where places may be, but given the pressure on school places generally, the answer is likely to be to apply for schools and see what happens as a definitive answer can't be given unless a formal application is made.
In Essex, schools which are their own admission authority handle mid-year applications themselves, so for the ones on your 'wishlist', you would need to submit an application through ECC for Springfield (a Community school), but to the other two schools individually.
Each school to which you apply must inform the LA admissions team that an application has been received, and whether or not a place has been offered. (Obviously, the team know the outcome of any applications made to schools through them).
If none of the schools you apply for have a place in the relevant year group, Essex CC will find your DS a place once you have provided the required proof of address.
www.essex.gov.uk/schools-and-learning/schools/admissions/move-different-school
This booklet has school details for the coming academic year - you actually want the details for the 2024 / 2025 academic year for a mid-year application, but most of them won't have changed.
www.essex.gov.uk/sites/default/files/2024-10/Primary%20Education%20Policy%20Directory%202025%20to%202026%20-%20Mid%20-%2018.10.2024.pdf