Nginx configuration for a sudirectory with PHP is returning 404 – Apache

I have a website www.example.com using a custom PHP framework and a CMS that loads a completely different framework. It was initially on apache2 and now I am trying to move to nginx. The www domain loads file but when it comes to loading the codebase in the subdirectory, it is simply returning 404 on my nginx. If I switch back to apache2, it loads fine. Here is my config which I basically cobbled from looking at other stackoverflow answers:

server {
listen 80 default;
client_max_body_size 256M;
access_log /var/log/nginx/site.access.log;

root /site;
index index.php;

if (!-e $request_filename) {
    rewrite ^.*$ /index.php last;
}

location ~ .php$ {
    fastcgi_pass php-fpm:9000;
    fastcgi_index index.php;
    fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
    fastcgi_param PHP_VALUE "error_log=/var/log/nginx/site_php_errors.log";
    fastcgi_buffers 32 32k;
    fastcgi_buffer_size 64k;
    include fastcgi_params;
}

location /cms {
    root /cms/;
}

location ~ /cms/.+.php$ {
    fastcgi_pass php-fpm:9000;
    fastcgi_index index.php;
    fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
    fastcgi_param PHP_VALUE "error_log=/var/log/nginx/site_php_errors.log";
    fastcgi_buffers 32 32k;
    fastcgi_buffer_size 64k;
    include fastcgi_params;
}

}

I tried this answer but did not have much luck. I must have some basic configuration incorrectly setup but can’t seem to figure out which. The .htaccess in the /cms is as follows:

RewriteEngine On
RewriteBase /cms

RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^(.*)$ index.php?request=$1 [L,QSA]

Edit: Updated the config using this example and I now can login to the CMS but it immediately goes into a 302 redirect loop which I can’t figure out why it is happening:

server {
listen 80 default_server;

# client_max_body_size 256M;

root /application;

index index.php;

server_name _;

# access_log /var/log/nginx/application.access.log;

location / {
   try_files $uri $uri/ /index.php$is_args$args;
}

location /cms {
        alias /application/cms;
        try_files $uri $uri/ @cms;

        location ~ .php$ {
            include fastcgi_params;
            fastcgi_param SCRIPT_FILENAME $request_filename;
            fastcgi_pass php-fpm:9000;
        }
    }

location @cms {
    rewrite /cms/(.*)$ /cms/index.php?/$1 last;
}

location ~ .php$ {
    include fastcgi_params;
    fastcgi_pass php-fpm:9000;
    fastcgi_index index.php;
    fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
    fastcgi_param PHP_VALUE "error_log=/var/log/nginx/application_php_errors.log";
    fastcgi_buffers 32 32k;
    fastcgi_buffer_size 64k;
}

}