invented to keep out the cold
They certainly were. Ileosa was the obvious next person to sit on the throne, so she might not be a conspiracy member — if the conspiracy failed, she would be expected. Conversely, that makes her better motivated to get Eodred killed, and she did claim the throne in the vision where the conspiracy went as planned. Tencednil probably won her bet, in other words.
Sweeping for Urgathoans is obvious, correctly gains her legitimacy she deeply needs right now, and will let whatever guards Ileosa has suborned plant evidence on anyone she wants to execute. Best to watch for claimed executions, then. It's possible there are some actual Urgathoans as well — that one wizard did manage to teleport out, for instance — but it's important to track who the conspiracy is killing. That will be the best sign for when they need to run.
Monsters from the sewers are... strange. The Vaults were opened by the Urgathoans, and the plague was very conveniently timed to give Ileosa legitimacy when she crushes it. Especially if that alchemist was right, and they actually can create a cure, which would be a masterstroke for gaining the trust of Korvosa. That all holds together very nicely, but why would the conspiracy be waking up monsters they'll have to crush? Things rising up from beneath are actually dangerous! Unless Ileosa is getting substantial funding, and probably magical support, from her coconspirators, she's not going to have the forces to handle them! Are they really dumb enough to not think of that? This is a lot of wizardry bent to a task with at least some subtlety, surely they considered the consequences at least a little...
Humans.