As for what "theoretical framework" is needed, you can think of the independence theorems as statements about finite sequences of formulas, so the "metatheory" doesn't need to assume the existence of any uncountable sets.
In more detail: first, assign different natural numbers to each formula (via Gödel coding), and then define a formal proof to be a finite sequence of such numbers where each one encodes a formula that follows logically from the previously-encoded formulas. Consistency means that that the formula $x \neq x$ is not logically derivable in any finite number of steps. This converts "ZFC is consistent" into a combinatorial statement about finite sequences of natural numbers.
With this reductive approach, you don't have to worry about assuming the existence of large infinite sets a priori. In fact, I think that all the relative consistency results gotten using Cohen's forcing technique could, in principle, be formalized just in Peano arithmetic.
Kunen's Set Theory: An Introduction to Independence Proofs contains nice clear discussions of these foundational issues.