I forgot to check to see if the 1.2.0416 update was installed before installing the unofficial patch. I know you're supposed to do them in reverse. Do I need to reinstall the UOP, or something more drastic (like reinstall the game)?

As far as I know, there is no problem installing them in another order. Just apply the 1.2.0416 patch. Make sure to use the correct patch (base game or SI version).

If your uninstall process will put everything back the way it was (depends not only on what tool you are using (Wrye Bash is your best bet; OBMM doesn't restore files), but also if that tool has the original files to work with), then that should be all you need. However, if you aren't certain it will restore everything, you are gambling. If you are out of luck, then you will just have problems later.

It sounds like you are just starting your install, so I would recommend playing it safe and re-installing. Problems never get easier to fix over time; only worse. It may not be what you want to hear, but that is the reality of it.

Edit: Patches are looking for particular sequences of bytes to locate where they start to make changes. If things aren't the way they are expected, they typically do nothing. (How it responds depends upon how much 'smarts' the patcher was written to handle.) So simply patching over another patch is unlikely to come out correctly unless they are both affecting completely different areas. But do YOU really know that? If not, then, as I said, it's a gamble.

-Dubious-