Officially, the African leg of the Arab slave trade ended in 1923 when Ethiopia, one of the last countries where slavery was legal, definitely outlawed it after years of incremental legislation against the practice. This was done by the Ras Tafari (later known as Emperor Haile Selassie) in order to gain admittance to the League of Nations.
Of course, the slave trade continued after that point. Inside Ethiopia, due to social & economic resistance to its abolition, people remained enslaved due to the financial investments slaves represented, as well as the limited reach of the Ethiopian government. (At the time, the Ethiopian government relied on the help of local aristocrats to exercise its power, many of whom were slave owners. Nevertheless, progressive elements in Ethiopian society persisted in freeing slaves & limiting the trade.) Slavery in Ethiopia only came to a definitive end under the Italian occupation (1936-1942). It must be emphasized that the Italian Fascists did not end slavery for altruistic reasons, but to weaken Ethiopian traditional society & strengthen their hold on the country.
Outside of Ethiopia, the trade continued much longer. Despite control of the major ports, the English & Italians could not stop every small boat that crossed the Red Sea with kidnapped people destined for the slave markets of the Arabian peninsula. Evidence of this comes from one Western tourist, Rosita Forbes, who writes:
When I was in Yemen in 1923, Abyssinian slaves fetched a high price
and the harems of the Tehama were full of girls who could not speak a
word of Arabic, so they could not have been long away from their
native Ethiopia. (cited in Richard Pankhurst, Economic History of
Ethiopia (Addis Ababa, 1968), p. 127)
For further information, Pankhurst devotes an entire chapter to the Ethiopian portion of the Arab slave trade, & Harold G. Marcus Haile Sellassie I: The Formative Years (1892-1936) (Lawrenceville: Red Sea Press, 1996) discusses the politics behind Ethiopia's abolition of slavery.