First, a little context: there were indeed quite a number of academics called to work at Bletchley, including a language-specialist friend of Tolkien named Angus McIntosh. So, the idea that Tolkien would have also gone to Bletchley is not absurd.

Angus McIntosh is famous for being Tolkien's squash partner; Tolkien suffered an ankle injury from a January 1936 match which took him off his feet. This led Mcintosh to later claim he was entitled royalties on The Hobbit and Lord of the Rings; note this is clearly a joke, and that The Hobbit was essentially in existence by 1936 and The Lord of the Rings wouldn't be started until 1937.

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In January 1939, Tolkien was asked if he was prepared to work in the cryptographical department of the Foreign Office, and agreed; he took a "training course" in late March.

In September, he wrote to Philip Unwin (one of his publishers) that he had agreed to start by spring and could still be called into service at any time.

In October, he was informed his services would not be presently required.

Finally, on the 19th of December, he wrote to Stanley Unwin (another of his publishers) that he had "the virtual headship of a department in this bewildered university" and "I am uncommandeered still myself, and shall now probably remain so, as there is (as yet) far too much to do here, and I have lost both my chief assistant and his understudy".

In other words, by December, as he was essentially left alone in charge of the English department at Oxford, it's likely his job situation is why he never was called into service.

One final note: the article mentions he was "keen" to join. In all likelihood, the marking of "keen" next to his name on the GCHQ document was simply meant as a guide on how to pronounce his name.

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Heims, N. (2004). J.R.R. Tolkien. Philadelphia, PA: Chelsea House Publishers.

Tolkien, J. R. R., Carpenter, H., & Tolkien, C. (2012). The letters of J.R.R. Tolkien. London: Harper Collins.

Scull, C., & Hammond, W. G. (2018, April 2). Addenda and Corrigenda to The J.R.R. Tolkien Companion and Guide (2006) Vol. 1: Chronology · Arranged by Date. Retrieved April 4, 2020, from https://www.hammondandscull.com/addenda/chronology_by_date.html