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We give a relative-consistency argument using a <finite-function collapse to countable size>. The assumed consistency implies that <ZFC> is consistent. Passing to the <constructible universe> gives consistency of $\mathsf{ZFC}+V=L$, which also satisfies the <Generalized continuum hypothesis>. Work in this ground theory, let $\delta=\omega_1^L$, and force with finite <partial functions> $\omega\rightharpoonup\delta$, ordered by reverse inclusion.
The union of a <generic filter> $H$ is a total surjection $h:\omega\to\delta$: prescribing a new domain coordinate is dense, and putting any specified $\gamma<\delta$ into the range is dense. Hence $\delta$ becomes countable. Set forcing preserves ordinals, and <absoluteness of constructible levels> implies that the extension has the same constructible universe as the ground model. Its $\omega_1^L$ is therefore still the old $\delta$, so it satisfies the stated <countability of constructible omega-one> condition.
We must also verify GCH after the collapse. The order has ground-model size $\delta$, hence the $\delta^+$-chain condition. By <cardinal preservation by chain-condition forcing>, all old cardinals at least $\delta^+$ survive. Every old ordinal below $\delta^+$ has size at most $\delta$ and becomes countable, so the old $\delta^+$ is exactly the new $\omega_1$.
A name for a subset of a fixed ground-model ordinal $\lambda$ can be chosen as a set of pairs $(\check\xi,p)$ with $\xi<\lambda$ and $p$ a condition: use an antichain deciding membership at each coordinate. Thus the number of such names in the ground model is at most $2^{|\lambda\times\mathbb P|}$. For subsets of $\omega$ this is $2^\delta=\delta^+$ by ground-model GCH. In the extension it gives $2^{\aleph_0}\leq\delta^+=\aleph_1$, and <Cantor theorem> gives the reverse lower bound. For every old cardinal $\lambda\geq\delta^+$ the bound is
$$
2^{\lambda\cdot\delta}=2^\lambda=\lambda^+
$$
in the ground model. Both $\lambda$ and $\lambda^+$ remain cardinals, so again the upper bound and <Cantor theorem> give the extension's equality $2^\lambda=\lambda^+$. These are all its uncountable cardinals. This proves <GCH preservation by a finite-function collapse> in the required case.
The forcing relative-consistency theorem now yields
$$
\boxed{\operatorname{Con}(\mathsf{ZFC}+\mathrm{NC})\ \Longrightarrow\ \operatorname{Con}(\mathsf{ZFC}+\mathrm{NC}+\mathrm{GCH}).}
$$
The construction in fact only needs consistency of <ZFC>. This argument uses the formal inner-model and forcing consistency theorems; it does not infer the existence of a <countable transitive model> merely from consistency.
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