If it's any consolation that's a very common problem on "factory" ukes, most of which have molded plastic nuts that are never touched by human hands. That's why we always recommend purchasing from a dedicated uke dealer who sets them up - one of the things involved in a setup is making sure that things like that are taken care of.
But, since you have the uke and need it to be right...there are two things to look at.
First, those appear to be Aquila strings, which are fairly "fat." You could try some fluorocarbon strings. Fluorocarbon is more dense so the strings tend to be a little thinner for a given pitch and tension. Often, just putting fluorocarbon strings on will make the C string fit the slot better (though once the string is in the slot the slot may still need lowering to be really right - that applies to all the strings).
If you like the strings that are on it, it's time to break out some very fine (320 grit or finer) wet-or-dry sandpaper. Using something very thin but stiff (like a small knife blade) to support the sandpaper, carefully work the sides of the slot a little at a time. Try not to let the sandpaper touch the bottom of the slot too much because until the string is seating you don't know if you need to remove material there or not. If the bottom of the slot is lowered too much the string will buzz when played open.
Or, if you have a good ukulele dealer nearby, you could take it to them. It's about a two minute job for someone with the correct files.
Oh, and welcome to UU, btw.
John