This is really an absurd problem to have. It's supposed to be that each country follows a calendar according to the culture there, which may or may not be different than its neighbours'. The Church there, of course uses the local calendar, and has its own local liturgical customs. Obviously if you went to another country, you would follow the calendar of that country. But when we have 4 churches on the same block, one following the old calendar, one following the new calendar, one following the Coptic calendar, one following the Armenian calendar, all as if they were in their home countries, and not in the country their in, it causes these silly practical problems. But they're not insurmountable problems. Even within the EO they can have two churches on the same street, one old calendar and one new, so people who might visit one or the other have this problem.
It depends on your needs, and what is best for you. If you occasionally visit there, but mostly follow the Coptic calendar at home and when you can go to Liturgy, I would start and stop fasting according to the Coptic calendar. But if you're mostly attending the community there, and going to the Coptic Liturgy rarely when it is available, then you might start and stop fasting according to their calendar. It isn't the end of the world if you celebrate the Virgin's feast with them while you are still fasting. Not ideal, but manageable. Ideally you would make the decision about when to start and stop fasting under the guidance of a father in confession, but if that is not possible, then the guiding principle is that fasting should teach obedience, so we should just avoid the temptation of suiting the schedule to our preferences and justifying it with various calendars. If we avoid that temptation, then which calendar it's done on doesn't really matter, in my opinion.
I know one person who is sometimes here, sometimes in India, and sometimes in England for the Nativity. If here or in India, it is on Jan7, if in England, it is Dec 25 with the BOC, yet he keeps fasting till the 7th because his heart is elsewhere. Not ideal, but it works, so don't get to hung up on what the right way is.