What influence does British immigration have on British society?

As there is no mention of time in your question, I can only discuss it in the background of18-19th century, because the scale and number of British immigrants at that time were enormous, and they were accompanied by British colonial expansion.

First of all, it is certain that British immigrants have strengthened the relationship between the host country and Britain, spread British culture, expanded British influence, and had a far-reaching impact on British politics, economy, culture, religion and diplomacy.

Secondly, let's talk about politics first, and maintain and consolidate the political system. The Queen of England and Britain have high honors in Commonwealth countries. If the system is changed, it will also be a blow to Commonwealth countries, and the control of immigrants over Commonwealth countries is obvious (such as the governors of individual regions). Second, promote economic development and close economic ties. The United States is an immigrant society, and there are many immigrants from Britain. There are British shares in American banks. Some conspiracy theorists believe that British bankers led to the American Civil War and the assassination of the president, and the establishment of colonies also made the industrial revolution flourish. Third, the popularity of English is the embodiment of culture. Fourth, the Puritans went to the North American colonies and established the Anglican Church in English society. Fifth, the Commonwealth and the United States are close to Britain in diplomacy, so it can be said that immigration has played an important role.