The good question is what part should be cut loss and what part should be kept, the British Empire was not viable in a liberal democratic world order and neither was it in communist world order, a surviving empire would be a mega-version of Apartheid South Africa and neither the British general population nor colonial subjects was willing to accept that. Of course all these thing is in no WW2 timeline or a timeline where Singapore holds.
So let start with pierce which need to be cut: India, Nigeria, East Africa, Southern Africa, Ghana.
So what should the British keep, the dominions outside South Africa are pretty obvious. But outside that Britain should focus on low population area with high value resources. So establish a continued British protectorate over the Persian Gulf states. I would suggest unite South Yemen, Oman, UAE, Dubai, Bahrain and Kuwait into a UAE style federation with the British monarch as president (similar to the emir of Abu Dhabi being president of UAE). This gives the British greater control over the global oil production. Do the same thing with Malaysia. Beside that keep control over strategic important position, so keep Zanzibar a protectorate (use the Zanzibar Genocide as excuse to intervene) and place South Asian refugees from East Africa on the island, this give the British base to intervene in East Africa. Keep Singapore, Suez, Aden (maybe as a British possession in the Greater UAE), and reach a agreement to extend the lease over Hong Kong. Beside that establish a informal empire over the weaker states much as the French did with that post-colonial empire.
If the British empire loses India, I think most would characterize that as it having "fallen." To me, this means that the liberal democratic world order needs to be butterflied away entirely, which requires at least a pre-WW2 POD.
You bring up settler colonies, which makes me think, why did white settlers never colonize the tropics in large numbers? The answers, of course, are (1) long travel times and general remoteness, (2) malaria and other tropical diseases, (3) high population density of natives. If these 3 factors can be mitigated, then that would open up the whole Empire for white settlement and likely allow British rule in these areas to be more deeply entrenched.
(1) will be solved on its own with the advances in air travel starting in the 1960s. (2) can already be solved by antimalarial drugs and insecticides by 1940. (3) is the biggest issue; as long as Europeans and natives live near each other in a colonial setting, there are likely to be attacks against the former, which is not conducive to colonization.
Britain was highly stratified by class before WWII, and every European country held varying degrees of white supremacist views, which does lend itself well to a colonialist system. Let's say that after World War 1, the colonial powers go more Imperialist as a whole, including Britain, France, Spain, Portugal, and the Low Countries. They end up in a war against the USSR; the US goes isolationist and stays neutral in the conflict, supports the Soviets for political reasons (Let's say Britain caused a fuss in South America again and the Americans are still mad about it).
The war grinds to a stalemate; without resources and manufactured products from the US, the Imperial faction goes harder on industrializing their colonies in order to make up the difference. They manage to set up a system where nouveau-riche natives who follow European customs are set up as a native "Aristocracy" and are rewarded with a decent portion of the trade income.
At some point, the war ends. The Imperial faction creates a UN-like body whose goals are (1) preventing war between its members, and (2) suppressing native rebellions (call it "enforcing the rule of civilized law worldwide"). The prevailing ideology becomes not a liberal-democratic one, but an imperial-democratic one: Certain rights and freedoms are guaranteed, but the default system is un-egalitarian, stratified by class, race, and geography. However, by necessity it is significantly more liberal than the previous imperial system, which keeps Mau Mau-type rebellions from spreading too much.
By the 1970s, advances in air travel and medicine allow whites to travel to colonies en masse. Large parts of the colonies are set aside for whites in an Apartheid-like system. There is virtually no visa restriction on travel within the empire for citizens of the metropole, but of course there exists a complicated tiered citizenship system for non-Europeans. I think that this system would be stable as long as it remains profitable; if it becomes a point of national pride to avoid the fate of Spain and the Ottoman Empire, then it may become politically verboten to advocate for the independence of colonies, especially as whites continue to move there.
Non-European countries in this system would probably develop economically faster than in OTL, and avoid the larger civil wars, but at the cost of extremely high inequality in terms of both socioeconomic status and human rights. More powerful European cities such as London or Paris would probably resemble Dubai in that imported guest workers would handle all the menial jobs under poor conditions for low pay.
It's unlikely that this global system would be as homogenous as I portray it here; certainly, the liberal-democratic world order that exists in OTL has not resulted in the entire world becoming fully liberal and democratic; quite the opposite, most countries pretend to be democracies but operate corrupt or despotic regimes. I expect something analogous to come about here: A large plurality of colonies probably present themselves as fully-fledged members of the Empire, but in reality have little imperial oversight nor authority outside the capital cities.
In this TL, the existence of the United States and Latin American countries would be counterpoints to this imperial system, since they would be independent nations not beholden to a European power. I'm sure that this new imperial ideology could be bent in such a way to accomodate them (For example, perhaps the Latin American countries "count" as European due to their cultural background) but something must be done about the US's rhetoric of liberty and equality.