Anglo-Indians are left-overs from mixed marriages in the colonial days - they are Indian people who generally speak Hindi, but have European names and are mainly Christians.
The polite name for an Indian is Indian. An Indian is not ethnic to Britain because, despite common usage, "ethnic" means "born in a place".
There are three states of British citizenship which apply to those from former British colonies; one is that the person has all the rights of a native Briton (and therefore is a citizen of the EEC), the second is that the person has rights to take up residence in Britain (but not the EEC),the third is that the person has British citizenship but no right of residence in Britain.
What people have to go through to take up residence in Britain depends on which residence status they have. A child born in Britain is not automatically a British citizen, and its residence status depends on the status of its parents. In many cases, such a child is in a position to choose its own nationality - I know of two brothers born in Birmingham, one of whom is British, and the other Ghanaian.