Hi,
The idea is to set up writing permission to NFS share for the owner of Apache and PHP processes on your web servers. To be successful with such scenario, it has to be the same user (the same uid) on every single web server in your configuration. Luckily, your environment seems to be homogenic, so we can be optimistic :) However, you need to check some things to be 100% sure. First check what a user is behind Apache/PHP processes:

ps -efjH | grep -i -e apache -e php

Output
root       10892       1   10892   10892     php-fpm: master process (/etc/php/7.4/fpm/php-fpm.conf)
www-data   10903   10892   10892   10892       php-fpm: pool www
www-data   10904   10892   10892   10892       php-fpm: pool www
root       11074       1   11074   11074     /usr/sbin/apache2 -k start
www-data   11075   11074   11074   11074       /usr/sbin/apache2 -k start
www-data   11076   11074   11074   11074       /usr/sbin/apache2 -k start

I believe it is www-data in your case, since you use Ubuntu distro. It might be another user if you changed Apache’s default configuration.
Though you can see a root user in some lines of the result, the processes which effectively work are their child processes, having www-data user privileges.

The next thing you need to check is the user id of www-data. Even if a user name is the same across the hosts, a user id may differ. You can find uid in passwd file. Run below command:

sudo cat /etc/passwd | grep www-data

Output
www-data:x:33:33:www-data:/var/www:/usr/sbin/nologin

Uid of www-data is 33. This user belongs to its own user group www-data only.You should get the same result on each host in your configuration.
Now, you need to change an ownership of your NFS share. You will be able to do that locally in the system that hosts this share. This system has to contain www-data user in its configuration too.

sudo chown -R www-data:www-data /var/nfs-share/

Use -R parameter to apply the settings to all subdirectories and files. Substitute /var/nfs-share/ with your actual NFS share.
Just in case, you could propagate appropriate rights across the share:

sudo chmod -R 755 /var/nfs-share/

Now you can mount NFS share under a subdirectory in the www root directory on web server. PHP app should be able to write to it now.