This sounds like Look To The West.
Link?
By the way, just to showcase how the TL I described above differs demographically from OTL, assuming South Africa is not in the UK:
Total population: 320 million. OTL UK+IE+US+CA+AU+NZ+Caribbean Anglosphere: 460; more than 100% of the difference is in North America. In 1900, this was 120 million, vs. 125 million in OTL.
Population by region: 75 million Britain (OTL 60), 10 million Ireland (OTL 6), 26 million Australia (OTL 25), 4 million New Zealand (OTL 4), 4 million Caribbean (OTL 5), 191 million North America (OTL 355) of which 25 are north of the Great Lakes/49th parallel (OTL 35) and 166 are south (OTL 320).
GDP per capita: £9,000. £1 = $5 in OTL 2015 PPP rates, so the average is a hair lower than in OTL.
Black population: 30 million. OTL: about 48 million. The big difference in population in North America comes from not having a long mid-20c baby boom, rather than from different migration patterns, so it would also affect the African-American population. Blacks are somewhat wanked in the sense that the income gap with whites is smaller than in OTL, and the early history of the police was more influenced by the metropole and less by slave catchers, resulting in less police brutality in than OTL.
Francophone population: 8 million. OTL: 7 million in Canada. With no American Revolution, Anglophones didn't settle the north shore of Lake Ontario to block French expansion.
Native American population: 10 million. OTL: 4 million between the US and Canada. US indigenous birth rates are much lower than those of any other US racial group, so between the much less genocidal policy in the ATL and the lack of a white and black baby boom, the indigenous proportion would be a bit higher than in OTL's Canada.
Hispanic population: 25 million identifying as having Spanish ancestry. OTL: 50 million. Mexico only borders the UK briefly, near the Pacific, and is also much richer. (In OTL, net migration from Mexico to the US turned negative this year, because of the strong Mexican economy.)
Asian population: 27 million. OTL: 30 million. Asian migration to the Anglosphere is recent, so the numbers are similar. They're a bit lower since British North America has much less of the Pacific coast.
Arab population: 5 million. OTL: 3 million. A smaller, more British North America consumes less oil than in OTL in the second half of the 20c. Mexico, which was a big exporter in the early 20c, is still richer than in OTL, but the Arab Gulf states are poorer than in OTL, so more Arabs emigrate to the UK.
Major metropolitan areas:
London 20 million. OTL 14 - in this TL, London is the political, financial, and cultural capital of a larger country.
New York 16 million. OTL 23 including New Haven and Allentown. New York got to export slave-farmed cotton for 30 fewer years, and is the financial and cultural capital of English North America but not of the UK.
Philadelphia 14 million. OTL 6.5. OTL's Philadelphia lost British investment in the late 18c since it was viewed as a center of revolution, unlike Loyalist-controlled New York. In the ATL, Philadelphia remained larger than New York for longer, so a few of New York's largest-city functions stayed there.
Boston 13 million. OTL 7, including Providence and Worcester. OTL's Boston was viewed as a center of revolution, like Philadelphia. The ATL's Boston also benefits more from being closer to Britain than in OTL.
Chicago 10 million. OTL 9.5. The Midwest is a bit smaller, but there are no major competitors on the Mississippi - St. Louis is French, Minneapolis is small, Kansas City doesn't exist.
Detroit 8 million. OTL 5.5. Detroit is this TL's Silicon Valley.
Atlanta 7 million. OTL 6. No major changes; it's a mid-19c outpost in the South build to help it industrialize. It's a bit bigger because the South has a bit more people.
Vancouver 6 million. OTL 2.5. Vancouver is the main gateway to the Pacific, taking in the role of Seattle, San Francisco, and Los Angeles.
Liverpool-Manchester 6 million. OTL 4.5, but these are really two separate regions in OTL. Liverpool gains more than in OTL from being closer to Ireland and to North America.
Cleveland 5 million. OTL 3. Much more a screw than a wank - Cleveland is Ohio's capital, and the main center of the auto industry, with all the post-industrial decline that characterizes OTL's Cleveland, Detroit, Manchester, and Birmingham.
Montreal 5 million. OTL 3.5. It's the capital of a larger province - the Quebec-Huron (*Ontario) boundary is much further west.
Sydney 5 million. OTL 5. The major difference is that Australia got less immigration in the 19c (more of North America was available for British emigrants) and more in the second half of the 20c (no UK membership in the European Community means the metropole's tariffs still protect Australia and New Zealand's farm exports).
Melbourne 4 million. OTL 4. See above on Sydney.
New Birmingham 4 million. OTL 1. New Birmingham was founded at the same time as Atlanta, so it was a bit bigger than in OTL in the late 19c; in the 20c, it grew at the same pace as Atlanta, because the racist politics that doomed it in OTL's second half of the 20c did not really happen.
Dublin 4 million. OTL 2. Capital of an Ireland with almost twice OTL's population and a kinder history, with earlier industrialization.
Birmingham 4 million. OTL 3.5. Slightly more immigration into Britain's larger cities - a large portion of the difference with OTL is black Caribbean.
Notable screws: Toronto does not exist. Washington does not exist, but Georgetown became a metro area of 2 million. Miami is a city of 2.5 million. *Seattle is a city of 2 million, and exists as a cheaper alternative to Vancouver. Minneapolis has 1 million people (the UK bought the west side of the Upper Mississippi from France to facilitate railroad construction).