Trump mother did not retain UK citizenship. Her oath of citizenship required her to renounce her UK citizenship.
I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty of whom or which I have heretofore been a subject or citizen; that I will support and defend the Constitution and laws of the United States of America against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I will bear arms on behalf of the United States when required by the law; that I will perform noncombatant service in the Armed Forces of the United States when required by the law; that I will perform work of national importance under civilian direction when required by the law; and that I take this obligation freely without any mental reservation or purpose of evasion; so help me God.
Some countries take the U.S. oath as revoking their citizenship. Not the U.K. The U.K., like the U.S., is cool with dual citizenship.
In the U.S., naturalization in another country does not end U.S. citizenship. Period.
In the U.K., the issue is discretionary. But, if the U.K. Secretary of State revokes U.K. citizenship, you can appeal. The U.K. Secretary of State did not revoke the citizenship of Donald Trump’s mother. Hence, she was a dual citizen. By lineage, so are her children. Donald Trump is a dual-citizen of the U.S. and the U.K. (this has been dormant and, so, isn’t an issue). Furthermore, because the Donald has established a meaningful tie with the U.K. via his major investment in Scotland, he has invigorated his (long dormant) U.K. citizenship, and his children could claim to be dual citizens. (The transfer of citizenship of persons born overseas only goes one generation. If your foreign-born parent doesn’t establish a relation with the home country, citizenship by lineage ends with him.) The only non-dual citizen person in the Trump household is Melania, because Slovenia is very restrictive about dual-citizenship. Donald Trump has major properties in Panama, Brazil, Turkey and the United Arab Emirates, in addition to Scotland. He is a very international person, with friends, business partners and investments all over the country and all over the world, including Hispanics and Muslims. I frankly think this is all very positive.
“People with dual nationality who are British nationals can be deprived of their British citizenship if the Secretary of State is satisfied that ‘deprivation is conducive to the public good’; there is a right of appeal.”