There may not have been a movement in Britain, but there were certainly individual left-wing anti-colonial intellectuals from the British colonies who wrote works in this vein. C. L. R. James from Trinidad was one, recognized even today for Black Jacobins, a history of the Haitian revolution published in 1938.
This event (contemporary to the French Revolution) seems to have inspired anti-colonials, whether they wrote in French or in English – Césaire also wrote a history of it.
James’s first version of the story was a play, apparently, and he later wrote the first novel by a West Indian published in the UK, according to Wiki. It might be instructive to compare his biography to Césaire’s. After the war Césaire managed to become the mayor of the capital of Martinique, and he was instrumental in setting up the arrangement where the island remained a département of France on its own terms. James was more of a loner, his leftist politics never achieved anything concrete for Trinidad, and he ended up mainly writing about cricket!
Concerning the Négritude movement, it’s worth remembering that its founders were not only anti-colonial activists, they were also poets and novelists who wanted to gain recognition as such not at home in the colony, but in the metropole (Paris). French writers, especially on the left, were more likely to form movements at that time (the 1930s and 40s) – would André Breton, leader of the Surrealists, have recognized Césaire and supported his poetic efforts without the calling card of Négritude? Perhaps not.
In Britain in those days, on the other hand, literary movements were unusual – only Pound’s Imagism and Wyndham Lewis’s Vorticism come to mind, and these were both on the right politically. So for James or his friends, founding such a movement might have been rather pointless. I think it’s this, more than the different ways Britain and France managed their colonies although there may also be linkages in this regard, that answers your question.