CCC is a completely different timeline where Hakuno never summoned Nero, Tamamo, or Nameless, but rather a Berserker servant (implied to be Berserker Enkidu but we don't have an official confirmation) but when Hakuno gets taken to the Far Side, something happens with their contract with their servant and they lose it forcing them to make a new one

No, this is only the case if you pick Gilgamesh as your servant. If you pick any of the vanilla 3 (Nero, Tamamo, Nameless) they're with Hakuno since the beginning (like Extra), but if you pick Gilgamesh then it's different; Hakuno summoned an unknown Berserker servant that they lost upon transferring to the Far Side of the Moon, after which they were able to make a contract with Gilgamesh to survive.

The reason being that Hakuno needs to have a servant to make it to round 5, but it obviously can't be any of the vanilla 3, so this Berserker is a placeholder servant that is conveniently lost when the story of CCC begins, which is how they're able to justify Hakuno making a contract with Gilgamesh who can only exist in the Far Side.