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I'd give yourself a little more time. Especially for production servers, I don't like to patch them for at least a couple of weeks because the updates could potentially screw something up and I want to ensure that I have time to do the updates, test, and make sure I have time to restore if necessary.
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I normally push the updates to clients the day after patch Tuesday. It's never caused me a problem though I know some will differ. In over 15 years I can only recall one patch in which I actually had to uninstall due to the problems it created. Though some patches can be buggy.
Servers I wait a few days and will usually update them within a week unless it's a patch for Internet Explorer. No web surfing on a server so not an urgent issue.