So I just updated SMIM (Static Mesh Improvement Mod) and now when I try to start my game I get the error message that objects are missing. Can anyone tell me what I may have done wrong, or if this is fixable?

This is almost always from a mod that is missing. Since you updated the mod, it may be the mod name for the new version is what is confusing things.

Load the game, make a forked save (new manual slot) and play on, it should be OK if this is the only change you made. Also, look in the readme for the mod to see if there are any notes about the Update and what to do.

The Nexus is down for maintenance (again! ) so I can't check the update, but if you're using the DLC, it's possible that Brumbek has either merged the dragonborn tern fix and furniture snow fix into the main esp file, or has removed the esp altogether in the latest version.

So it shouldn't be a problem then if I play with it in this state?

I just checked, and the esp files appear to be the same - did you select the same install options when updating? Try reinstalling the mod, making sure you choose the same install options as you did last time you installed (eg. select dragonborn tern fix if you had that fix before).