Wordpress - How can I get a list of users by their role?

There may be some different way to do that, but most proper way to do that is following.

<?php

$args = array(
    'role'    => 'Your desired role goes here.',
    'orderby' => 'user_nicename',
    'order'   => 'ASC'
);
$users = get_users( $args );

echo '<ul>';
foreach ( $users as $user ) {
    echo '<li>' . esc_html( $user->display_name ) . '[' . esc_html( $user->user_email ) . ']</li>';
}
echo '</ul>';

?>

Here the simple approach to grouping roles.

$wp_roles = wp_roles();
$result   = count_users();

foreach ( $result['avail_roles'] as $role => $count )
{
    if ( 0 == $count )
        continue; //pass role none

    $args = array(
        'role' => $role
    );

    $users = get_users( $args );
    $user  = array();
    for ( $i = 0; $i < $count ; $i++ )
        $user[] = esc_html( $users[ $i ]->display_name ); //show display name

    //output
    echo wp_sprintf( '<h2>%1$s</h2><ul><li>%2$s</li></ul>',
        esc_html( $wp_roles->role_names[ $role ] ),
        implode( '</li><li>', $user )
    );
}

When you find users with Ultimate Member Plugin Roles, You have to add "um_" to your role value. For example, you created the role name "Client" in Ultimate Membership Plugin, then $args would be

$args = array(
    'role'    => 'um_client',
    'orderby' => 'user_nicename',
    'order'   => 'ASC'
);