Personally, I don't think that on the day of independence (if it ever happens), the English in Scotland and vice versa, will suddenly feel out of place, and needing to negotiate their foreignness. They'll presumably still have the same jobs, houses, families, friends. The same goes to people living in 'their' country, but knowing people who've migrated from over the border. For me, this suggests that naive views about nationalism and cosmopolitanism that take the national border as a determining factor are misguided. The border is a historical and social construct that may or may not be an accurate reflection of the lives of people, and certainly isn't the only determining factor.
One question I have is whether someone from Glasgow, living in London, should be considered to be in any way cosmopolitan or part of a multicultural society. For it seems to me that the both words are loaded: the first pertaining only to the elite, and the second only to people with darker skins. The Scottish are neither... but perhaps we should be thinking about their difference too. Furthermore, given that the people/cultural forms post-independence are just the same as now, shouldn't we consider movement to London (internal migration) as creating multiculturalism too.
That said, I think that the idea of the border also sets us up to think of those who do the moving as making the multiculturalism or being the cosmopolitan. However, these ideas have to be relational: just like the movement of objects in space, movement of A towards the stationary B and vice versa are equivalent. Of course, one person moving from Moscow to Manchester will become a lone Russian surrounded by Mancunians. But if lots move to Manchester, then populate a street where one Mancunian lives, then in some respects it is as though the Mancunian had moved to Moscow. It's thus possible to be cosmopolitan or living multiculturally, having stayed still but having had the world to move in.
Indeed, the term cosmopolitanism is often associated with the kind of people who have gone to work overseas and then later return. However, they are often doing just as the Muscovites mentioned before: they are in particular enclaves, sending their kids to international schools, and being with other expats as opposed to 'locals'. This looks a bit like the ghettoisation associated with 'ethnic others' coming from the other side of the world, but because it's the elite doing it, it's judged very differently. I'm not convinced it really is cosmopolitanism: real cosmopolitanism can be seen in the urban kids (including 'locals') who have friends from everywhere, and just live with it, without making a fuss, and without engaging the state... but often they're not the kind of cosmopolitans the elite wants.