https://aviation-safety.net/investigation/cvr/transcripts/cvr_af4590.php shows that AFR4590 didn’t use heavy in its call sign even though Concorde had a MTOW greater than 300,000 lbs. Unlike BAW, AFR didn’t have a specific Concorde callsign, so why did they not say heavy:
The post When is 'heavy' used in callsign? shows that the usage of "Heavy" depends on the country.
The general ICAO usage is to say "Heavy" only in the initial message with a frequency.
That usage applies to France, where the linked CVR transcipt took place (no differences are listed in the French AIP). Since that transcript does not show the initial contact with tower, it doesn't show when "Heavy" was first used.
USA however deviates from this standard, and this difference from ICAO is listed in the USA AIP:
4.9.2 In the U.S., the word "heavy" is used in all communications with or about heavy jet aircraft in the terminal environment. In the en route environment, "heavy" is used in all communications with or about heavy jet aircraft with a terminal facility, when the en route center is providing approach control service, when the separation from a following aircraft may become less than five miles by approved procedure, and when issuing traffic advisories.
The general ICAO usage (4.9.2) can be found in ICAO Doc 4444:
4.9.2 Indication of heavy wake turbulence category
For aircraft in the heavy wake turbulence category the word "Heavy" shall be included immediately after the aircraft call sign in the initial radiotelephony contact between such aircraft and ATS units.