If I were in such a position, I would probably tell the students to pronounce every word properly and to feel comfortable and confident as long as native speakers can understand what they're saying.
Has anyone, who was not born and bred either in the UK or in the states, has learnt to speak in an American or British accent to such a point that people native to that country would take the person as one of them in terms of accent and pronounciation? I'm probably wrong. Some genieuses may have done so before. But I don't think this is an objective feasible for most people, esp adults, providing that they don't have luxury of time to remember the way every sound, word or sentence is said in a specific accent, as children do.
I've heard that many actors and actrees can quickly pick up an accent. Yet, even these people can't get it right all the time. Russle Crowe was pissed when Mark Lawson suggested, on a BBC Radio 4 programme, that there were tinges of Scottish accent in Robin Hood's voice, and left the scene. And they're prefessionals, people who're briliant at their native language and trained to do another accent.
Another thing is, there are so many American and British accents. People from Manchester speaks differently from those blokes of London. In gigantic cities, such as New York or London, people, I guess, don't pay much attention to try to find out if one grows up locally, for there are probably as many accents as they can get. But I seriously doubt, say, after a person spends a couple of months in a lesson, people from any part of the UK, after a short conversation or a while, will not think that this guy is, at least, not from their area, accentwise.
After all, one who has invested lots of money and time will very likely end up speaking an American or British accent, which no one in the states or the UK would do. If the purpose of learning a foreign language is to communicate with people, why should some one pay much attention to speak like a person from a particular part of the UK or the states, letting alone the fact that it's nearly a mission impossible.
Say, I am Chinese. There are thousands of accents in China. My father, who was a Chinese language teaher before his retirement, forbad me from speaking my local accent when I was a kid. I can't do it at all. And I don't have an accent that can help people identify where I am from. Two years ago, I was recognised as a non-local guy in a real estite agent in my hometown, for I couldn't speak my local dialect and had hints of southern accents, after I had spent roughly a third of my life in the opposite part of China. But, people in the south think I have a northern accent. I learnt Beijing dialect from Telly but people there believe my accent is from a neighbouring province. I am just lucky to belong to the half of Chinese people whom most people can understand.
So my point is that having a specific accent is not that important. What's more important is the way one expresses themselves and whether one delivers their words properly, pronounciationwise.