There are a lot of stages between throwing a few nukes (which has been done before) and an all-out-everybody-who-has-nukes free-for-all global nuclear war. If two or three tactical nukes are detonated without any nation totally panicking, but taking a moment to wait and see, the war is likely to stop. That will happen if nothing is ever seen coming over an ocean in any direction. Nobody is completely crazy, and a lot of people in those chains of command want to live.
Remember the world went through a couple of decades of hundreds (maybe thousands) of nuclear tests in the ocean, on land, and even in space. So, nuclear detonations 'way over thataway, even a number of them, don't spell the end of the world over here.
Whatever exchanges occur, moreover, will be in the Northern Hemisphere, and the deadliest radiation decays quite rapidly.
When we were working out the plans for national "enduring deterrence" and putting those plans into place, we figured that a distance of 250 miles from the nearest detonation and 60 days of food and water (which actually requires quite a surprising lot of water when you include necessary washing and sewage...not so much food) with some very careful sanitation makes survival very, very probably.
Then the best long-term bet is getting down to South America as quickly as possible.