It appears to be an idiosyncratic reading which spread after D&D's release.
While I haven't read every possible miniature wargame, sources suggest that the dice notation was an invention of Dungeons & Dragons after its release, and that the varying "dee" and "die" pronunciations arose thereafter.
According to Playing at the World, a book which details the history of D&D, most wargames before D&D used either readily available six-sided dice, or twenty-sided dice marked 0-9 and used for percentages (e.g. Tractics (1971)). Full sets of polyhedral dice (d4, d6, d8, d12 and d20, but not d10) were sold in 1973, and their use was pioneered by Original D&D (1974), but the dice notation was not yet used.
The specific term "die (number)" was used in the original D&D to refer to a result of a roll, not a die type. For example, the bandits listing in Monsters & Treasure (1974), p.5:
If there are over 200 bandits there will be a 50% chance for a Magic-User (die 1-4 = 10th level, die 5-6 = 11th level) and a 25% chance for a Cleric of the 8th level.
TSR's Fight in the Skies (1975) (video) used percentile dice, but not the "d20" type notation. The need for such notation was likely derived from the high variety of dice used by D&D. "Fight in the Skies" says in places "roll two dice", without notation. Even the AD&D Monster Manual (1977) did not yet use dice notation.
The earliest use of dice notation that I'm aware of is Dragon #7 (June 1977), What To Do When the Dog Eats Your Dice, by Omar Kwalish, an alias of Tim Kask. Here he refers to six-sided dice as "D6", printing a table from Fight in the Skies to allow d6s to simulate percentile dice (suggesting that polyhedral dice were still uncommon). In his recent YouTube video series, Kask does not use the pronunciation "die", suggesting perhaps that it was never standard pronunciation at TSR.
The Player's Handbook (1978) introduces dice notation, but does not define a pronunciation. For example:
The cleric has an eight-sided die (d8) per level to determine how many hit points (q.v.) he or she has.
The AD&D 1st edition Dungeon Masters Guide (1979) uses the term "die (X)" in two tables: on page 172, referring to a result of 6 or 7 on the die; and page 206, where "die 20" refers (presumably) to a roll of 20 on the die, i.e. a critical hit. We might infer from this that Gygax himself did not pronounce "d20" as "die 20", otherwise the term "die 20" would not make sense here.
However, the pronunciation "die 20" is used by Len Lakofka, early D&D contributor who gamed with Gygax and had influence on AD&D, though he did not work directly at TSR as Gygax and Kask did. "Die 20" doesn't appear in any official work, but it does appear in Lakofka's self-published free work Devilspawn (2009), pp.98-110:
Roll six die four and bring them in based on that die roll.
In the event of an encounter, roll a second time on die 20 and consult the table below:
In other words, it's likely that Lakofka is among the earliest influential inviduals to pronounce "d" as "die", for which it is ultimately an abbreviation. Similar idiosyncratic pronunciations of words sometimes occur in the D&D community, particularly pre-Internet, when one has only seen obscure terminology written, and only heard it spoken within the D&D community.
My educated guess is that it arose at some point during the development of AD&D between 1976 and 1979, perhaps linked to Lakofka's involvement or arising independently in the D&D community, and spread by word of mouth despite not being formally defined in any official TSR sourcebook.
For more information from sources who were there at the time, you could contact Tim Kask, who has a good chance of responding to comments on his YouTube channel (such as on his latest video); and on the Dragonsfoot Forums, an OD&D forum where Len Lakofka himself sometimes posts.