How do I find out if my sudoer privilege timed out?

Use the option -n to check whether you still have privileges; from man sudo:

-n, --non-interactive

Avoid prompting the user for input of any kind. If a password is required for the command to run, sudo will display an error message and exit.

For example,

sudo -n true 2>/dev/null && echo Privileges active || echo Privileges inactive

Be aware that it is possible for the privileges to expire between checking with sudo -n true and actually using them. You may want to try directly with sudo -n command... and in case of failure display a message and possibly retry running sudo interactively.

Edit: See also ruakh's comment below.

Run:

sudo -nv

If your sudo privileges have timed out, this will exit with an exit code of 1 and output:

sudo: a password is required

If you have valid cached credentials, this command will succeed and output nothing.

So, to put it all together, here's a scriptlet that will silently check if you have valid cached credentials:

if sudo -nv 2>/dev/null; then
  echo "no sudo password required"
else
  echo "sudo password expired"
fi

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As other answers/comments mentioned, the -v option ("validate") to sudo silently renews cached credentials if there are any or else prompts for authentication in order to generate cached credentials, and the -n option ("non-interactive") prevents sudo from generating any interactive prompts, such as the authentication prompt.