I don't mean just in France, England, Spain, Portugal, etc, but how did people come to know about the new world in less involved parts of the old one like the Ottoman Empire and China? How did nations bordering the Atlantic ocean react? Was there ever any worry of invasion from the western seaboard during colonization?
I've never heard of any fear of invasion, and word spread very slowly even within Europe (where much more is written on this topic).
A good place to start is Hugh Honour's The New Golden Land: European images of America from the discoveries to the present time. London: Allen Lane, 1976.
It is clear that one can find nodal points for the dissemination of the information (beginning with Spain and Genoa and moving outward from those places). Nations with standing ambassadors at the court of Spain found out sooner. Nations who sent staff for their ambassadors (mapmakers, painters, etc.) got even more information, as they could copy the paintings (and later, frescoes) wherein the New World was described and depicted.
We know that the discovery of the New World initially caused barely a ripple in African nations bordering the Atlantic. Trade went on as usual (remember, the Spanish and English explorers had a hard time finding anything in the New World to trade until the tobacco thing).