As far as we know, the Children were native to Westeros. There's no connection to the rest of the world. That said, the White Walkers are both intelligent and hateful. If they wanted to, they could use boats. Or possibly just freeze the Narrow Sea and march over, depending on the AoE of their freezing aura.

I'm digging it so far, and don't mind most of the deviations from the source material. Tulip and Cassidy have both been set up in such a way that their backstories from the comic won't work the same, which is the main point that concerns me. Their introduction scenes were both incredibly cool, but I'm worried the show…

I remember back in the days when LiveJournal was relevant, someone actually produced a play based on Titor's stories of the future called Time Traveller 00. I wasn't able to see it, but there are photos somewhere. Interesting costume design.