There is no reliable solution. Anything wget can do, your browser can do too.
Your best shot is something like sending to wget to http://example.com/mysript.php?accesskey=some+really+obscure+password+that+only+you+should+know
and check for that access key in your script. Of course, if anyone gets the password, this kind of protection is useless, but it's a far more consistent point of failure than blindly relying on User-Agent sniffing.