I think the problem came from Rewrite mod. Removing the 3 .htaccess files allowed me to access the app urls.
By the way, I tried on a fresh clean Apache server with the following .htaccess files and it worked :
/var/www/cakephp/.htaccess
<IfModule mod_rewrite.c>
RewriteEngine on
RewriteBase /cakephp/
RewriteRule ^$ app/webroot/ [L]
RewriteRule (.*) app/webroot/$1 [L]
</IfModule>
/var/www/cakephp/app/.htaccess
<IfModule mod_rewrite.c>
RewriteEngine on
RewriteBase /cakephp/app/
RewriteRule ^$ webroot/ [L]
RewriteRule (.*) webroot/$1 [L]
</IfModule>
/var/www/cakephp/app/webroot/.htaccess
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /cakephp/app/webroot/
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule ^(.*)$ index.php/$1 [QSA,L]
</IfModule>