Technically speaking there is only one Islam, as it is a universal and global faith. But Islam is always, and has always been, understood, lived and practised in very local contexts. Muslim traditions have evolved with local nuances in mind. From China to Morocco to the USA, the practical application of Islam feels different. Indeed it should be rooted in its local ‘urf (customs).
So when we talk of a ‘British Islam’, it is shorthand for seeking the nuances in a British approach to practicing Islam, just as there can be an Indian, Pakistani or Egyptian approach.