There were several families:
descendants of Princess Beatrice and prince Henry of Battenberg and those of her niece Victoria of Hesse and his brother prince Louis of Battenberg. Only the line of Louis and Victoria continues in the male line in the UK.
Descendants of princess Mary Adelaide of Cambridge and Duke Francis of Teck. Their oldest son prince Adolphus of Teck became the first marquess of Cambridge. Their second son prince Francis of Teck died before WWI and the third son prince Alexander became the first Earl of Athlone and was married to princess Alice of Albany, cousin of George V (Alexander's brother in law through Mary of Teck) and Alice was the sister of Carl Eduard Duke of Coburg.
Princess Marie Louise and Helena Victoria of Schleswig-Holstein-Sonderburg-Augustenburg were the daughters of princess Helena and prince Christian. In 1917 when George V changed his dynasty's name from Saxony-Coburg-Gotha to Windsor, Teck became Cambridge and Battenberg Mountbatten the princesses remained princesses but without any land or area they went from HH princess of Schleswig-Holstein to HH princess.
More distant relations like the Gleichens did not change their name.
As for the titles in the UK the highest noble title is Duke but their status in society was such that you needed to be very wealthy to live up to what was expected of a Duke so the title of marquess was chosen as it carried less of a financial burden.
Louis Battenberg became Marquess of Milford Haven (a sea port)
His nephew Marquess of Carisbrooke
Adolphus Teck Marquess of Cambridge and his younger brother only had one surviving son and a daughter so became Earl of Athlone.
As for their status, both the Battenbergs and Tecks were morganatic branches of reigning German dynasties. Battenberg of Hesse und by Rhine and Teck of Wurttemberg. Their status in Germany was that of neither meat or fish but in the UK Queen Victoria was more welcoming.
Victoria Eugenia, Alice and Louise of Battenberg all married into reigning dynasties but they were only acceptable Royal brides because of their maternal line. Their mothers Beatrice of the UK and Victoria of Hesse und by Rhine were full blown Royals and that made the girls suitable brides. The same applied to Mary of Teck. Her mother made her a great granddaughter of George III. The fact that she was a morganaut did not matter in the UK but that she was not a foreign bride with political links to another country than Great-Britain was seen as a positive attribute.
Were the Bettenburgs the only noble/royal families switching their allegiances from Germany to the United Kingdom. Did many need to change their names? When the Battenburgs came to the UK and swore their allegiance to the king, did they receive titles of a lesser rank that those they enjoyed using in Germany?
Of course, all of the above had long been domiciled in Britain and were effectively part of the British establishment. Yes, they lost their German titles and Prince Louis of Battenberg lost his naval role but otherwise their lives in Britain largely carried on as before.
The Tecks had been resident in Britain since 1866 and, apart from time abroad on British military service and a brief period in Italy from 1883 to 1885, Francis, Duke of Teck, never lived in Germany after the 1860s. All four of his children were born in Britain and never lived in Germany at any stage of their lives.
The two princesses of Schleswig-Holstein were born in Britain in the 1870s and brought up there. True, Princess Marie Louise did live in Germany after 1891 when she married Prince Aribert of Anhalt but after that disastrous marriage was annulled in 1900, she spent the rest of her life in Britain.
Prince Louis of Battenberg enrolled in the British naval in 1864 at the age of 14. His naval career lasted for more than 40 years and he became First Sea Lord in 1912. Although two of his children were born in Germany, he had been domiciled in Britain for 50 years when World War I broke out. He was effectively British but his German title and ancestry – and ill-conceived public prejudice - forced him to resign in November 1914. King George V expressed his confidence in Louis by making him a Privy Councillor. His four children – Alice (who married Prince Andrew of Greece in 1903 and became the mother of Prince Philip – she was resident in Greece in 1914), Louise (who married King Gustav VI Adolf of Sweden in 1923), George (Marquess of Milford Haven, who married a Russian countess) and Louis (Earl Mountbatten) did spend holidays in Hessen, Germany as children – and both Alice and Louise spent time in Darmstadt - but were otherwise effectively British.
The Battenberg children of Queen Victoria’s daughter Beatrice were all born (at either Windsor or Balmoral) between 1885 and 1891 and were brought up in Britain, never living in Germany at any time.
The three younger Gleichens, a morganatic branch of the princely house of Hohenlohe-Langenburg, were all born in London in the 1860s and spent their lives in Britain, apart from Count (later Lord) Edward Gleichen’s distinguished military career in the British army in Africa and as British military attaché in Berlin and later Washington DC.
It would, I believe, have been a surprise if they had not given up their German names during World War I, when anti-German feeling was rife. They were all effectively British rather than German anyway and had been so for decades. None could in any sense be seen as "leaving Germany".