If you could find a direct link to the specs of the JSON format that UE4 uses for importing tiles it would make the whole task much easier — I’ve looked at the documentation you linked above, but it’s rather huge to sift through, and seems more a user manual, which might not necessary contain these specs.
The video from YouTube clearly shows that Tiled map editor supports exporting to that JSON format. Since Tiled is open source, as a last resort one might peek at the source code to gain info on how the tile image are serialized to JSON:
The project is quite big and contains sources in C++, Java, Python and Lua (since plugins seem to be codeable in various languages). Probably the export to JSON functionality is to be found somewhere inside the
src/plugins/ folder (
src/plugins/json1/ being good candidates).
Needless to say, this involves much more work than reading the specs of the JSON format required by UE4, which are language-agnostic and don’t require unraveling the code. As a last resource, you could open an Issue on Tiled repository and ask its author where to find those specs (he, or whoever wrote the plugin should be able to indicate the specs doc to you).
Once you have the specs, it’s a matter of working out how to create a plugin (DLL) that can export to UE4 via JSON. This should be pretty doable, unless you need to use True Colour (since the current plugin interface is not documented for True Color, and seems to be buggy too).