What you are asking, essentially, is that the people of the 15th and 16th century develop disease-fighting techniques that were essentially unavailable till the 20th century. There is no way that you are going to inoculate millions of people in order to prevent them from dying when they come into contact with diseases they've never encountered before. Even if it's not a combination of smallpox and measles, it could be a combination of typhus and smallpox, plague and cholera, influenza and cholera, typhus and influenza, influenza and plague, etc. And how do you expect the people of the Antilles or Mesoamerica to treat disease they've never seen before.
Actually I was thinking of only inoculating the Europeans going over to the Americas. I was thinking that if Europeans had knowledge of Germ Theory they would become hesitant about people moving across borders possibly bringing disease with them so when vaccination becomes possible(hopefully earlier than OTL)sailors, travelers, soldiers and the like would be vaccinated as a way of protecting themselves from future problems or even their own people as they have no idea whether these diseases are already in the Americas.
As far as Typhus, Influenza and Cholera are concerned all of these ailments respond well to hydration therapy(carrot soup, or in the case of the Americas tomato soup possibly)which has been known for millenia in India.