I would say not to expect an autonomous, let alone autocephalous, British jurisdiction for some time. North and Central America have over 50 bishops - certainly more than enough for a united autonomous/autocephalous synod - and the closest we have is the Orthodox Church in America which is not recognized as autocephalous by most Churches (though, the majority of Orthodox Christians are under Churches recognizing it) and it only has 34,000 attendees (compared to several times that for the Greeks) on an average Sunday.
As for whether or not people baptized by the Greeks consider themselves Greek Orthodox, by the Russians Russian Orthodox, etc., I can only tell you based on those people I know. Let me start with this, at the Greek Orthodox Church in Prescott, AZ (St. George Orthodox Church - IIRC that is exactly what the sign says, no mention of "Greek") there are two priests. One of them got a theological degree from St. Herman's Seminary (an OCA seminary in Alaska, and following the Russian tradition), and is a Greek priest - though not ethnically Greek. The other is of Irish-Catholic heritage and was trained through St. Stephen's Course (this may have been because he may well be the oldest non-monastic ordained in North America in the history of Orthdooxy) and is actually a priest of the Romanian Archdiocese, and is a supply priest for them but usually serves alongside the main priest at the parish, and serves on his own on Friday mornings. So, those are two examples of how inter-connected Orthodoxy can be in America (the UK may well differ). But, of all the people I've met at St. George, I have never heard anyone refer to themselves as Greek Orthodox, just Orthodox. They don't talk about Russian Orthodoxy or Greek Orthodoxy or Romanian Orthodoxy, but about Orthodoxy (unless people are comparing the liturgical, or other unique, practices of a particular ethnic group, then of course they mention the ethnic affiliation).