[Houshalter]

I said almost everyone, obviously you aren't (likely) related to remote islanders. However you'd be surprised how much mixing there is between population. All you need is one individual from another population to mix with another and within a few generations he is everyone's ancestors.

I'm not certain how much mixing there was between the new and the old world. However it did happen a lot and it's likely a large number of people do have some new and old world ancestors.

[maxerickson]

You are overstating the amount of mixing between populations (and somewhat discounting the mixing within populations).

1000 years ago, people in villages and towns were regularly marrying their (2,3,4) cousins and rarely marrying people from 'different lands'.

Another way to look at it: for 1 person to become the ancestor of just several thousand people takes 'a few generations'. Getting to whole populations takes lots of generations.