In this paper, we present a new solution to the problem of large-scale multi-building and multi-floor indoor localization based on linked neural networks, where each neural network is dedicated to a sub-problem and trained under a hierarchical stage-wise training framework. When the measured data from sensors have a hierarchical representation as in multi-building and multi-floor indoor localization, it is important to exploit the hierarchical nature in data processing to provide a scalable solution. In this regard, the hierarchical stage-wise training framework extends the original stage-wise training framework to the case of multiple linked networks by training a lower-hierarchy network based on the prior knowledge gained from the training of higher-hierarchy networks. The experimental results with the publicly-available UJIIndoorLoc multi-building and multi-floor Wi-Fi RSSI fingerprint database demonstrate that the linked neural networks trained under the proposed hierarchical stage-wise training framework can achieve a three-dimensional localization error of 8.19 m, which, to the best of the authors' knowledge, is the most accurate result ever obtained for neural network-based models trained and evaluated with the full datasets of the UJIIndoorLoc database, and that, when applied to a model based on hierarchical convolutional neural networks, the proposed training framework can also significantly reduce the three-dimensional localization error from 11.78 m to 8.71 m.