Q: Does anyone know any more information about this theorem? Does it have a name?

This result is simply referred to as "Meier's theorem" in the literature. It follows from the property that for large $n$ the chi-square distribution tends to a narrow Gaussian, $$f_{n}(x)\mapsto \sqrt{\frac{n}{4\pi}}e^{-(n/4)(x-1)^2}.$$ Now you can expand the function $R$ in a Taylor series around the peak of the Gaussian, $$R(x_1,x_2,\ldots x_k)=R(1,1,\ldots 1)+\sum_{i=1}^k \left.(x_i-1)\frac{\partial R}{\partial x_i}\right|_{(1,\dots,1)}+\frac{1}{2}\sum_{i=1}^k (x_i-1)^2 \left.\frac{\partial^2 R}{\partial x_i^2}\right|_{(1,\dots,1)}+{\cal O}(x-1)^3,$$ and perform the Gaussian average, $$\mathbb{E}[R]=R(1,1,\ldots 1)+\sum_{i=1}^k \left.\frac{1}{n_i}\frac{\partial^2 R}{\partial x_i^2}\right|_{(1,\dots,1)}+{\cal O}(n^{-2}).$$

Q: Does anyone know any more information about this theorem? Does it have a name? Why is it true?

This result is simply referred to as "Meier's theorem" in the literature. It follows from the property that for large $n$ the chi-square distribution tends to a narrow Gaussian, $$f_{n}(x)\mapsto \sqrt{\frac{n}{4\pi}}e^{-(n/4)(x-1)^2}.$$ Now you can expand the function $R$ in a Taylor series around the peak of the Gaussian, $$R(x_1,x_2,\ldots x_k)=R(1,1,\ldots 1)+\sum_{i=1}^k \left.(x_i-1)\frac{\partial R}{\partial x_i}\right|_{(1,\dots,1)}+\frac{1}{2}\sum_{i=1}^k (x_i-1)^2 \left.\frac{\partial^2 R}{\partial x_i^2}\right|_{(1,\dots,1)}+{\cal O}(x-1)^3,$$ and perform the Gaussian average, $$\mathbb{E}[R]=R(1,1,\ldots 1)+\sum_{i=1}^k \left.\frac{1}{n_i}\frac{\partial^2 R}{\partial x_i^2}\right|_{(1,\dots,1)}+{\cal O}(n^{-2}).$$