Seems to be at least plausibly true, both in the fact of the euthenasia, and that the timing was at least partially to meet the Times deadline. Ultimately sourced to the Dr's diary.

https://books.google.com/books?id=xUa-P74h-nEC&pg=PA93

At about 11 o'clock it was evident that the last stage might endure for many hours, unknown to the patient but little comporting with the dignity and serenity which he so richly merited and which demanded a brief final scene. Hours of waiting just for the mechanical end when all that is really life has departed only exhausts the onlookers and keeps them so strained that they cannot avail themselves of the solace of thought, communion or prayer. I therefore decided to determine the end and injected (myself) morphia gr.3/4 and shortly afterwards cocaine gr. 1 into the distended jugular vein

The determination of the time of death of the King's body had another object in view, viz. the importance of the death receiving its first announcement in the morning papers, rather than the less appropriate field of the evening journals

Per this LA Times article, the information flowed from the diary, through historian Francis Watson http://articles.latimes.com/1986-11-28/news/mn-15949_1_mercy-killing

The issue arose in an article written for the magazine History Today by British historian Francis Watson. He said the 70-year-old king's doctor administered separate injections of morphine and cocaine about 40 minutes before George V died on Jan. 20, 1936.

Watson, 79, had published a biography of Dawson in 1950, five years after Dawson's death. Watson said he had not mentioned the lethal injection at the time because he did not want to embarrass the physician's relatives or the Royal Family.

http://www.thecrownchronicles.co.uk/history/george-v-death-1936-doctors-helped-speed-up-his-death-injections-morphine-cocaine/

It was only in 1986, through Dawson’s biographer, that it was revealed Dawson administered two injections to the Monarch – fatal doses of morphine and cocaine – to speed his passing and make it painless. His notes said this was so that the announcement was made ‘in the morning papers rather than the less appropriate evening journals’.

Less than an hour after these two injections, George slipped away in his sleep.

The Dawson notes as they have been called, are now preserved in the archives of Windsor Castle; Francis Watson, the doctor’s biographer, looked at the notes five years after the doctor’s death in 1945. Lady Dawson requested that he omit any reference to the euthanasia.

Watson wrote a journal article for History Today in 1986, which included the information: “Perhaps I should have included it in the book [on Dawson’s life] at the time,” Mr Watson said. “Lady Dawson did not want it in the book and I quite readily agreed. I didn’t think it appropriate.”

http://www.bmj.com/content/308/6941/1445.1.full

His reputation would have been considerably diminished, however, had it been known that when the king was suffering from cardiorespiratory failure in January 1936 he administered a lethal combination of morphine and cocaine at a time when the king was already comatose and close to death. His action remained a well kept secret and the truth came to light only 50 years later when his private diary was opened, Dawson having died in 1945.

http://www.nytimes.com/1986/11/28/world/1936-secret-is-out-doctor-sped-george-v-s-death.html

As he lay comatose on his deathbed in 1936, King George V was injected with fatal doses of morphine and cocaine to assure him a painless death in time, according to his physician's notes, for the announcement to be carried ''in the morning papers rather than the less appropriate evening journals.''

The fact that the death of a reigning monarch had been medically hastened remained a secret for half a century until the publication today of the notes made at the time by Lord Dawson, the royal physician who recorded that he administered the two injections at about 11 o'clock on the night of Jan. 20, 1936

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