Question: Did John Gray cite any source for this claim? If so, what was it? I am seeing this excerpt tweeted quite a bit and it looks like there are endnotes/footnotes, but they are not included in the tweet and I do not own his book.

Posting whatever source Mr. Gray cited (if any) might help a person qualified to answer your question sink their teeth into it.

This is probably a badly interpreted mix-up of two separate events: disbandment of YuC Scouts in 1919, and the GPU's crackdown on non-pioneer scouting organizations on April 23, 1926 (two years after Lenin's death).

YuC Scouts (literally, Young Communist Scouts) were a short-lived organization, formed in 1918 of a number of pre-existing groups and clubs supporting the Bolsheviks. It had some financial support from several Soviet state authorities, including Vsevobuch (Universal Military Training), the People's Commissariat for Education and the People's Commissariat for Health. According to Vladimir Zotov (Из истории скаутского движения в России. 1909-1926 / From history of the Scouting Movement in Russia. 1908-1926), the disbandment was a result of pressure from the Komsomol (Young Communist League) trying to rebuild the YuC by adding "political instructors" and "political comittees" to the existing structure. Those political instructors were never as popular as actual scoutmasters, and the YuC scouts ceased to exist as a single entity in 1919, simply due to the lack of financing.

Without the support of Vsevobuch, the People's Commissariat for Education and the People's Commissariat for Health, the All-Russian YuC Union lacked facilities and resources. It crumbled away, but the scouts remained, and the Scouting Movement sprouted new roots.

There were several more short-lived organizations such as the Moscow Scout Team (1920-1921). Zotov mentions that Lenin himself visited a gathering of Moscow scouts in Autumn 1920, making a speech about "the role of youth in the building of new life". The Bolsheviks never gave up on the idea on rebuilding the Scouting Movement as a pro-Soviet, truly Communist organization. Nadezhda Krupskaya (Lenin's wife and major politician) published several books and articles about the scouts: in the article "РКСМ и бойскаутизм / All-Russian Young Communist League and Boy Scoutism" (1922) she criticized the idea of YuC Scouts and called the scouting movement itself a bourgeois entity, but praised the efficacy of scouts' methods of education and advocated for the Young Communist League and youth unions to apply these methods in practice.

The result was the establishment of Spartak Young Pioneers Organization, later renamed Vladimir Lenin All-Union Pioneer Organization. The founder of the pioneer organization, Innokenty Zhukov, was one of the leaders of the Russian Scouting Movement since 1915. Under Zhukov, the pioneers quickly become a massive, sprawling organization with semi-mandatory participation for school students and millions of members as a result; the organization existed until the fall of the Soviet Union in 1991.

Some scouting groups, claiming being apolitical, continued to exist until 1926, when GPU (the State Political Directorate, the Bolshevik not-too-secret police, which received judicial powers the same year) forcibly closed non-pioneer scouting organizations. According to Vsevolod Kuchin's book "Скауты России. 1909-2007" (Scouts of Russia. 1909-2007), 26 Moscow scoutmasters were arrested on April 23, 1926. The choice of the date was probably conscious: April 23 was the Saint George's Day (the Scouts' Day) according to the old Julian calendar (which was used in the Russian Empire until 1918).

26 scoutmasters, including 7 women, were arrested. In the autumn of the same year, they faced a trial before the GPU Judicial College which sentenced them to different terms of punishment, according to 1922 Criminal Code of the RSFSR in force at the time. Some were sentenced to 3 years of imprisonment in Solovki prison camp, and others to 3 years of exile to northern and sourthern regions of Russia.

Kuchin mentions that Vladimir Zotov, the author of the first book mentioned above, was among the Moscow scoutmasters the GPU arrested in 1926 and sent to Solovki prison camp. He survived and was released three years later, worked as a museum guide in Kaluga, participated in WWII as a Red Army soldier, but never allowed to return to Moscow until 1965; he died in 1978.

Some other scoutmasters were less lucky. Innokenty Kozhevnikov, a former Red Army commander, and Dmitry Shipchinsky, another Moscow scoutmaster, being prisoners themselves, worked with underage prisoners of GPU Labor Colony, another prison institution on the same island: the prison warden decided their skills would be useful in educating the youth. In 1929, Kozhevnikov and Shipchinsky escaped the prison camp and were hiding in a forest somewhere in the island for 15 days, receiving food from other prisoners and awaiting for a ship to escape. They failed to escape the island, and Shipchinsky was eventually captured and executed. (This story is told in page 371 of Kuchin's book with reference to memoirs of Dmitry Likhachev, a famous linguist and another prisoner of the same prison camp at the time).

In summary:

• the year 1919 probably refers to disbandment of a communist scout organization due to pressure from the Young Communist League and lack of financing, not a mass execution.
• there was an actual purge of non-communist scoutmasters (adults, not kids) in 1926, 2 years after Lenin's death.
• the scoutmasters were sentenced to imprisonment, not execution, but some of them were actually executed years later.

UPD: The source of the story of execution of Moscow boy scouts is indeed Sergei Melgunov's book "Красный террор в России: 1918-1923" (Red Terror in Russia 1918-1923), published in Paris in 1924. The only time when boy scouts are mentioned in the book is an author's footnote commenting an execution of several school students in Oryol in 1921. The footnote says:

Other shootings of the kind took place earlier. In 1919 some Boy Scouts were shot in Moscow, and in 1920 the same fate (for "espionage") was meted out to the members of a tennis club.

Melgunov does not provide any sources for this claim, and Zotov and Kuchin do not mention such shooting. The execution of some Boy Scout members in 1919 isn't impossible, but Melgunov never said that all the Boy Scouts of Moscow were executed.

For now, I'm assuming it's either conjecture or fabrication. Although it may be buried in a history book somewhere exactly as he describes it. The act itself doesn't surprise; it was 1919 Russia. But internet searches should produce an instance outside of Gray's book, I would think (Maybe one is yet to be revealed, we'll see). There's a whole gamut of mundane actions and words attributed to Lenin available, yet nothing on this very specific, shocking, easy-to-grasp, uniquely brutal act, something sensational enough to feed many thousands of anti-communist narratives. Hard to see how it got overlooked rather than played up.