Summary: We characterize those Tychonoff quasi-uniform spaces \((X,\mathcal {U})\) for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family \(\mathcal {K}_{0}(X)\) of nonempty compact subsets of  \(X\). We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space  \(X\) is uniformly locally compact on  \(\mathcal {K}_{0}(X)\) if and only if \(X\)  is paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show that a Hausdorff topological space is \(\sigma \)-compact if and only if its (lower) semicontinuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces  \((X,\mathcal {U})\) for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on  \(\mathcal {K}_{0}(X)\) is obtained.