Not really sure how to pull this off. At no point was the Monarchy as an institution unpopular enough that a good amount of the British population wanted to switch to a Republic. Yes there was high-strung republican sentiment during Queen Victoria's withdrawal after Prince Albert's death (1861-1871), which peaked with the establishment of the French Third Republic, but that wasn't a majority. The elite would never allow such a thing as it would destroy their own power.
Best bet I can think off would be the 1917-1918 bolshevik scare after the Russian Revolutions. George V was terrified that it would spread to Britain, which is one of the reasons he block the asylum offer to the Imperial Family and left his cousins to die. So maybe the workers of the UK take inspiration and stage their own revolution? IDK that's all I got.