The trick shown in the component fact table applies to your own hypercube queries as well: you can add the labels and language parameters.

In JSON or XML, this leads to a new Labels field that appears in each resulting fact. This is an object that associates every dimension, member and concept (and not only) with a label.

You can change the language (if available) with the language parameter.

You may notice that not only report elements are given labels: Entity IDs are replaced with a nice name, while dates and units are automatically formatted (in English only at the moment).