Hi,
I am seeing the same issue on the desktop site as well. Error 403 means, that the request cannot be completed because there is no permission to access the handler. In this case, the server cannot access the WordPress ajax handler. Therefore the search query is not even executed.
This is either because of a firewall policy, or a malformed .htaccess rule in the root directory. I suggest checking any custom .htaccess rules, firewall plugins, or any custom firewall/NAT rules.