This sounds to me like Tomorrow's Magic, originally published as two volumes, Winter of Magic's Return and 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.

This sounds to me like Tomorrow's Magic, originally published as two volumes, Winter of Magic's Return and 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+ years since I read it myself), but the focus is definitely on humans.

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.

This sounds to me like Tomorrow's Magic, originally published as two volumes, Winter of Magic's Return and 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.