We prove up-to-constants bounds on the two-point function (i.e., point-to-point connection probabilities) for critical long-range percolation on the $d$-dimensional hierarchical lattice. More precisely, we prove that if we connect each pair of points $x$ and $y$ by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$, where $0<α<d$ is fixed and $β\geq 0$ is a parameter, then the critical two-point function satisfies \[ \mathbb{P}_{β_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+α} \] for every pair of distinct points $x$ and $y$. We deduce in particular that the model has mean-field critical behaviour when $α<d/3$ and does not have mean-field critical behaviour when $α>d/3$.