Is it the right if an immigrant to GB in the last 200 years to say whether or not people in NI are British?The views of the British are irrelevant to most Unionists. They are British because they consider themselves to be so. They don't care what the so-called "real British" think. This is common enough among the descendants of colonial settlers, the pied-noirs in Algeria were exactly the same with regard to "mainland France". The Gibraltarians (albeit they are mostly descended from "native" Spaniards) are also comparable.
The covenanting tradition involves loyalty in exchange for defined guarantees. Don't forget that the "independent Ulster" strain of Loyalism has always been historically strong. True blue "British like Finchley" are a small minority among Ulster Unionists. Most are Unionist because they feel that it is in their community's interests to be part of the UK as opposed to a united Ireland. The idea that most Unionists will somehow give over centuries of hostility to the CNR community is wishful thinking. A process has started with Alliance-leaning Unionists becoming open to a united Ireland. This is where Nationalism needs to concentrate its efforts. The problem is that Sinn Féin is incapable of refraining from triumphalism and petty fights with the DUP because it enhances their electoral base.
Is it in our gift to say that those immigrants or their descendants are not British?
Are Irish Americans,English,Scottish etc not "Irish"?
Nice can of worms you've opened here.