Ants are quite resistent to mold and unless they actively avoid it it most likely doesn't pose a threat.
That little patch of mold is like nothing and it's probably just bacteria growing on the colony's toilet area. You can start worrying when the entire surface of the cotton is black but even then it's probably not an immediate danger.
The best thing you can do is to put a straw through the cotton (it MUST be large enough for the queen to pass), put the tube into an outworld (plastic container with Fluon or alcohol+babypowder as barrier is sufficient, but wash out the container before you use it) and offer a second tube. They're probably not gonna move though, their tube looks fairly clean.
This setup has the additional advantage that you can place food in the outworld (should be close to the nest) because feeding them inside the tube really disturbs the colony a lot.
It'd look like this.
(Straw was probably a bit too small but I couldn't immediately find a straw that fits a Camponotus queen)
Tubes (nest tube and replacement tube) each with straw and paper tent.
Here you can see them *** SPAM *** (mounted a small heat lamp above the outworld). I recommend cutting out a small part of the straws front so they have this sort of groove-like entrance area.
You will most likely notice when they are ready to move out, at least if they have more than a dozen workers. Mine started to congregate at the front of the tube for several days (the queen usually remained at the back end of the tube) before they moved.
They even dumped the brood into the outworld right before they moved.
Now that they're in the new nest I have trouble identifying the queen because in the last days there hatched a couple of majors that are just as big as her (here she's easy to spot though, right on top of all the brood).
Also, this is an impression of how messy a test tube can actually become.