Here is a way to view the mini that can put the mind of a TOS only person at ease:
Look at the mini like an alternate history. If we assume that the armistice at TOS was real, not a ruse than everything else in the mini is OK. The real Adama that day went on to complete his career. Every character on the Galactica that day has long since retired from Colonial service. In our own military most people retire after 20 or at most 30 years of service. A few Generals or Admirals might make up to 40 years. Since 40 years elapsed since the armistice and the start of the mini noone at the armistice is still in the fleet! The last name Adama may be fairly common in the colonial world. William Adama may be a distant relative of the Original Commander Adama but he is not him. His son Lee is not TOS Apollo or is his son Zak (a common name even in our society). The names are similar but I know a lot of Johns, James, etc.
So what do we have: All new people who are not the original TOS people in any way shape or form. Treat them as original people on an aging Battlestar Galactica 40 years after TOS in a world where the armistice did happen. The call sign "Apollo" and "Starbuck" may be as common as "Mavarick" or "Gunner" in our military.
Now the only issue is the origin of the cylons. Since this doesn't really impact the plot it doesn't matter anyway. Human-cylons can come along whether cylons started as man made objects or an alien made object.
I feel better already!