How widely accepted and used is the International Standard Atmosphere?
In most aviation contexts, the International Standard Atmosphere is used universally. This includes things like developing aircraft performance charts, weather forecasts, and flight planning. All flying rules and other aviation standards are all based on ISA.

There are several atmospheric standards. However, for the most part, all of these standards are the same up to about 32km altitude. Since most airliner traffic occurs between about 10km and 20km, there is no need to reference different standards in different locales. 

The International Standards Organization (ISO) publishes the atmospheric standard as ISO 2533:1975.  Sometimes this is referred to as the ISO Standard Atmosphere, which would have the same ISA initialization.  However, when used in textbooks, performance charts, flight planning, etc., the term ISA is always understood to mean "International Standard Atmosphere". For the rest of this answer, assume ISA to mean International Standard Atmosphere.

ISA is also an initialization of one of the other standards: ICAO Standard Atmosphere, published by the ICAO in Document 7488/3. The ICAO standard was first published following the 1950 ICAO Council, and underwent multiple changes over the years. in 1979, the Council approved the re-calculation of the data tables to align with the ISO standard of 1975. Once this was complete, the ICAO Standard Atmosphere and the ISA were identical. The ICAO document states that, "any future revisions of the ICAO Standard Atmosphere will be subject of the coordination of the two bodies [ICAO and ISO]". 

Because these two standards are the same, AND also have the same initialization of ISA, AND aviation standards world-wide are driven by ICAO, there are frequent mis-uses of the term ISA within the aviation world as being the ICAO standard. However, when used in the application of aviation "things", like calculating True Air Speed, knowing whether the temperature at an altitude is + or - from ISA; it is the International Standard Atmosphere that is meant by ISA.