In most cases, it took a very long time. The problem is climate - tropical fruits are not easy to grow in Europe (today, there is some cultivation in greenhouses). South American fruits and other crops rapidly spread through the tropics, to tropical Africa, India, and SE Asia. Cassava crossed the Atlantic to West Africa, and the Pacific to the Philippines, and rapidly became a very important crop in both Africa and SE Asia. The cashew was introduced to India by the Portuguese in the 16th century. The guava and passionfruit also followed these routes. The peanut rapidly spread through Africa (helped by pre-existing cultivation of the similar bambara nut) and crossed the Pacific to China where it became a major oilseed crop.
As far as Europe is concerned, the star among the South American crops was the potato. Unlike the others, it is far from being a tropical plant, growing in the Andes. The potato became popular in Europe in the 17th and 18th centuries - this makes it a slowly-adopted American crop. The potato, productive in cold Northern Europe, became a very important crop. The other place where the potato had a large impact was Tibet, where it allowed the population to triple. The introduction of the potato to Europe has been discussed before:
u/Jan_van_Bergen in https://www.reddit.com/r/AskHistorians/comments/3se3r5/did_frederick_the_great_antoineaugustin/
u/gothwalk in https://www.reddit.com/r/AskHistorians/comments/4ag9rk/how_did_potatoes_become_a_staple_in_european_diets/
u/unklethan in https://www.reddit.com/r/AskHistorians/comments/aawkkb/what_is_the_history_of_potatoes_like_where_did/
However, the tropical crops and fruits made little impact until modern transportation could bring them fresh to Europe, either from South America, or from places like India and SE Asia where they had been introduced to. The passionfruit is grown today in the Canaries, Maderia, and the Azores, and appears to have been introduced there early. If you consider these "Europe", then the passionfruit would be the first South American tropical fruit to be popularised in Europe (but only a limited part of Europe).
The above crops are the most important South American crops/foods that spread around the tropical world. (Other crops found in both North and South America also spread, and the ones that could easily be grown in Europe, such as maize, chillies, and Phaseolus beans, became important there.)
The brazil nut, much better suited for pre-modern export than tropical fruit, was being exported to Europe by the early 17th century, but only became popular in the 18th century.