Site moved to HTTPS and loads as HTTPS but permalinks show as http

If you are using Apache as your web server then add

# ensure https
RewriteCond %{HTTP:X-Forwarded-Proto} !https 
RewriteCond %{HTTPS} off
RewriteRule ^ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301]

in your .htaccess file, this can be found in the root wordpress directory.
Alternatively you can use WP Force SSL plugin.