Could Britain losing the First World War have ended or displaced the British monarchy?
Short answerOnly an extreme defeat that destroyed the Grand Fleet and enabled an invasion of Britain would directly endanger the royal family at home.
NotesDefeat abroad would not by itself place the British monarchy in immediate danger. Germany would first need to inflict a massive disaster on the Grand Fleet, removing the main barrier to invasion. An amphibious force would then have to overcome both the Royal Navy and the army retained in Britain before it could seize the country. Control of the surrounding seas is therefore the decisive condition in this counterfactual, rather than any British loss in the war automatically threatening the crown.