In both cases, the month was brought into existence through the advocacy of individual men, committed to elevating the names and histories of Black adults and children, past and present.
First, to the history of Black History Month in the United States. I've pulled information just about timing from an older response on the history of the month:
Unlike the other commemorative months brought to the attention of Congress by organizations and groups, African American History Month (also officially known as Black History Month) was shepherded into existence by the work of one Black historian, Dr. Carter G. Woodson. He announced the first Negro History Week in mid-February 1926, a week selected to hit the anniversary of Abraham Lincoln’s birth on February 12 and Frederick Douglass’ on February 14. The week was celebrated in various ways across the county for decades, culminating in a group of college students from Kent University advocating for federal recognition in 1970.
At this point, it's also worth highlighting another significant event in February related to Black American history: National Freedom Day. Richard R. Wright, a contemporary of Woodson, lobbied the federal government to recognize February 1 as a momentous day in American history. On the date in 1865, Lincoln signed the 13th Amendment, outlawing slavery in the United States.
Neither slavery nor involuntary servitude, except as a punishment for crime whereof the party shall have been duly convicted, shall exist within the United States, or any place subject to their jurisdiction.
The creation of National Freedom Day in 1948, campaigning by young people, and Black History Week lead to the formal recognition of Black History Month in 1975.
The recognized creator of Black History Month in the UK is Akyaaba Addai Sebo, a Ghanaian-born activist and member of the Greater London Council (GLC) who organized the first event in October 1987. In a recent interview, he was asked about the timing of the month. He offered:
There is historical link to Black History Month as celebrated in the US in February because of the inspiration of Dr. Carter G. Woodson who set it up there. We drank from the cup of Dr. Woodson but decided on a particular period of the year that will engage most the minds of children and youth in the UK. We settled on the propitious month of October when the weather was not cold and children were fresh after the long summer vacation and had less to worry about exams and tests and the camaraderie was stronger as they shared experiences. We believed that they would absorb more if their living environment buzzed with positive vibes, instructions and images about themselves and their origins, thus celebrating who they are as “Africans” who gave the world the concept of monotheism (the worship of a one and only God); who helped to install the first electric lighting system in London, Amsterdam and New York, in the person of Lewis Latimer, a pioneering partner of Thomas Edison and Nikola Tesler (credited with lightning the world) and many more. October in the UK as February in the US is to inculcate self-pride and especially in children. Self-pride is the catalyst for achievement and there is no greater “truth” than knowing yourself.
The initial celebration featured Dr. Maulana Karenga, creator of the pan-African and Black American holiday of Kwanzaa which further speaks to link between the American and British Black History months.
For more on the history of the month, this piece by Patrick Vernon, a British social commentator, provides additional context on the political events of the era that shaped the formation of the month. I wasn't able to find additional context on the Jubliee Proclamation he mentions so I'm putting the link down here.