What is the date of first use of the TAF format?
For the current format, it was 1993 when it was standardized.

On July 1, 1993, a new, revised Terminal Aerodrome Forecast (TAF) format and code replaced the existing TAF code. This new code is in effect in most countries. The exceptions are the United States and Canada. These two countries issue Terminal Forecasts for domestic use and TAFs for the military and those airports serving international aviation.
On January 1, 1996 the U.S. and Canada will convert entirely to the new TAF code. After that date, the new code for weather reports (METAR) and TAFs will be used worldwide. The North American code used in the FT (and SAO) will be discontinued. Although the new TAF code is being adopted worldwide, each country is allowed to make modifications or exceptions to the code for use in each particular country. The TAF code, as described here, is the one used in the United States for those airports serving international aviation and will also be used after the January 1, 1996 conversion date. The World Meteorological Organization's (WMO) publication No. 782 &quot;Aerodrome Reports and Forecasts&quot; contains the base METAR and TAF code as adopted by the WMO member countries. (Source)

noaa.gov also confirms the date for the US.
The WMO publication No. 782 can be found here.