Looking in Wikipedia and its sources, the Chambers Dictionary of Biography, and Oxford and Cambridge guides to literature, there is no reference to any honours received by Saki, even by his real-life name of HH Munro. Wikipedia does mention that "in recognition of his contribution to literature, a blue plaque has been affixed to a building in which Munro once lived on Mortimer Street in central London." A blue plaque is fixed to a British building or site in or at which a famous person lived or worked, or where a famous event took place. They are mostly in London, though there are similar schemes throughout the UK, and in the US and Australia. The plaque reads
"HECTOR HUGH MUNRO alias SAKI 1870-1916 Short Story Writer lived here"
It was erected by the cultural organisation English Heritage, which now runs the Blue Plaque scheme, in 2003. Here is an image:
The district of London in which you find Mortimer Street, Fitzrovia, was at the time Munro lived there a poorer and less Bohemian neighbourhood than it later became.
Perhaps the strongest recognition of his work is the influence it had on later British writers. They include: PG Wodehouse, who wrote the Jeeves and Wooster books and also worked as screenwriter and lyricist in the US, AA Milne, who wrote somewhat more than just the Winnie the Pooh books for children, and Noel Coward (whom some call "The Master"), playwright, lyricist, novelist and performer. Later writers with whom his work can be compared include Roald Dahl in his short stories, and Tibor Fischer's blackly humourous novels.
Saki is unlikely to have received any honours during his short life, as his writing was almost all pointedly satirical. He sits firmly in a strand of British writing which begins with gothic novels such as Mary Shelley's Frankenstein. He can be compared in style with contemporaries such as EF Benson, and in the US HP Lovecraft.