Directories where already set to 755, files to 644. It appears that some directories were owned by a different user, changing the wordpress directory and all subdirectories to www-data:www-data solved the problem. So this did the trick, for future reference:
sudo find wordpress -type d -exec chmod 755 {} \;
sudo find wordpress -type f -exec chmod 644 {} \;
sudo chown -R www-data:www-data wordpress