I was just looking for the same thing and found this VSCode extension:
move-ts
It is very young, but seems to be under active development. I tested it on my Windows machine and immediately ran into an issue with backslashes being used. There is, however, already a pull request to fix this (not by me).
As soon as this is fixed, I think this might be what you (we) are looking for.
EDIT: As of 2017-May-11, the windows issue is fixed.
There is an ongoing discussion for a “refactor move file” suggestion for Typescript itself here, but it seems to be very early stage.
It seems, until this is built, this functionality is unlikely to be included in VSCode itself, as this open ticket for VSCode explains, which tracks the main feature request you linked to in your edit.