Those .hgts files are in the same general format than the .hgt files, however they contain the elevation data in 4 bytes floating point numbers instead of 2 bytes integers.
Global Mapper won't recognize the .hgts extension. Therefore, when asked about the format of the data you have to choose the option "SRTM (Space Shuttle Topography Mission)".
But unfortunately that's not enough, because of the difference in the data type. Therefore you have to provide a header file (.hdr) that contain some information about how to interpret the data. Thankfully, the .hdr file is just an ascii text file, so you can create or edit it easily.
One way to get a template of the file is to download an standard .hdr file, open it in Global Mapper and then immediately export it in SRTM format. Along with the .hgt file, it will create a .hdr header file.
If you change the extension of the file to .txt and open it in a text editor it will look like this:
To adapt this header to work with hgts files we need to change the number of bits (NBITS) from 16 to 32, the total bytes per row and column from 7202 to 14404, and the NODATA value from -9999 to -32768.
Additionally, if you want to use the header for other tiles you can change the latitude and longitude of the upper left corner (ULXMAP and ULYMAP, that in this case is 50° West and 1° North).
In contrast to Global Mapper, QGIS doesn't automatically detect that the data type is float (i.e. decimal numbers), and might don't recognize the extension. To solve those problems you have to add an additional line to the header file:
And you might need to change the data file extensión to .bil (instead of .hgts).
I did a test and my .hdr file ended looking like this:
After those edits, if you rename the file back to .hdr (and optionally change the .hgts extension to just .hgt for Global Mapper or .bil for QGIS). You should be able to load the file without any problem. However, my version of QGIS renders it completely white, and I had to manually adjust the limits of the gray color scale to get it to display properly.
The examples above are for the file n00w050.hgts.zip (South America folder), so instead of exporting to get an .hdr example, it might just work if you copy the text above and paste it in a text file with the same name of the .hgts file, adjust the Lat/Lon of the upper left corner, rename to .hdr and that's it.
There are also scripts to automatically generate the .hdr files, as you see they are quite simple, and write such script wouldn't take much. This newer version of the SRTM (NASADEM), is still in a preliminary release, I describe it in this answer, and there I also mention a script to generate the .hdr files, you can give that script a try, it might solve your problem without messing with the .hdr header files.