How can I test a new nameserver before updating the registrar records?

If you're on a *nix system, you should be able to use either nslookup or dig. With both of those commands you can specify what server you would like to query. Simply use your new nameserver as the server. With dig, the query would look something like

dig @<your-nameserver> <hostname-to-look-up>

So, if you wanted to query google's public dns server and ask it about the address www.google.com, you'd use

dig @8.8.8.8 www.google.com

which, amongst other things, would tell you that www.google.com is an alias for www.l.google.com -- and then would give you a bunch of IP address (A records) for that name.

For both Windows and my remote Linux(CentOS) servers I use nslookup tool which works on both platforms:

nslookup new-sub.domain.com 8.8.8.8
// and for more details:
nslookup -debug new-sub.domain.com 8.8.8.8