The first thing that I did was zoomed out, surely Shakespeare wrote about something in the New World?

No. It is amazing how small his world was. He was born and grew to adulthood, in the world where Spanish dominance kept England from attempting to explore the world. While Jamestown was settled before he died, he never wrote about it.

I've updated my understanding of how (un)aware people were in this era of the larger world. I have no idea why I would have ever expected otherwise.

The map contains a bunch of references to America, the West Indies, Guiana, and Mexico. (Often with a connotation of "faraway exotic place" or "exciting new international development".)

He may not have written about the British colonies but the New World was clearly at least somewhat present in his mind and his audience's minds.