This sounds to me like Tomorrow's Magic, originally published as two volumes, Winter of Magic's Return and Tomorrow's Magic. https://www.goodreads.com/book/show/743391.Tomorrow_s_Magic

Not first person, but most of the other points check out based on my own memory and the Kindle sample:

 - Main character is Wellington "Welly" Jones, overweight and with poor eyesight.
 - Vision and weight issues prevented Welly from attending the Cardiff Military Academy; as it is he is getting a general education at Llandoylan School.
 - The Devastation 500 years before is discussed in the first chapter; the words "nuclear war" are used.
 - There is magic in the setting, but that is not clear to the characters early on--if you didn't finish the book, you may not have seen much of it.
 - Along with his best friend Heather and a boy who claims to be Merlin, Welly goes on a quest to find and bring back King Arthur.
 - The cover shown on Goodreads includes a fair amount of green and grey.
 - There may be some magical creatures lurking around the edges (it's been 30+ since I read it myself), but the focus is definitely on humans.