OK, Round up the usual suspects. Here are a few ideas grouped by arrival date:
The Thor Heyerdahl route. Natives of the Americas sail the Pacific, much in the same way the early Polynesians did, in the first millennium BC. They spread potatoes, new world curcubitae, peanuts and maize to their nerw island homes, but have only mythical memories of their origin in pre-Incan civilisations. First contact with Melanesian populations spreading west is made in Australian waters, and the crops are adopted in New Guinea by the time Gupta India and Qin China learn of them. They reach Egypt by trade routes through India and are there associated with that country, eventually becoming 'Indus beets', 'Taprobanean wheat' and 'Indian gourd'. The peanut is initially adopted only in China and spreads to India and Africa in the 1200s as 'Chinese almond'.
The Carthaginians sail the Atlantic in the quest for lucrative trade, but find little enough that is rewarding. However, when an expedition headed to the Azores is blow off course, they reach the Carribean and encounter a settlement of earlier shipwrecked sailors mixed with locals and subsisting on local crops. The Carthaginian seafarers decide to take all who want to come home with them, provision their ships for next year's navigation season, and depart. However, the trade winds leave then blown off course, their supplies nearly exhausted, on the African coast. Another settlement is made and the remaining seed stocks used for farming. It takes them three seasons of hard labour, peril privation, and canny trading to provision themselves for the journey back. The council of Carthage hears their reports, and find that this island is too hard to reach and not worth trading with. Their seed stock is readily accepted, though, and maize and sweet potatoes conquer North Africa. By 146 BC, there is a saying in Rome that 'No good came to Italy from the Punic but his wheat'. Cocoa beans will only become a trade item much later, when the seedlings planted by the Punic castaways in Central Africa have spread over the entire region.
The massive expedition commissioned by the Huangdi Emperor reaches the Americas, and some of the ships return with natives, animals, plants, herbs, and other rarities. However, as it turns out none of them are the sought-after elixir of immortality, they are consigned to their fate in the care of subordinate palace gardeners and menagerie keepers who neglect some, appropriate others, and sell a few. By the time the Qin dynasty falls, seedlings have become scattered all over China in the gardens of nobles and officials. That is why the chronicles later record that in the second year of the Han dynasty, a famine in the Yen province was alleviated by the exertions of official of the second rank Wang Tsu, who mandated the growing of a special kind of root local farmers called the 'Emperor's Garden Bread Radish'. Maize and peanuts by now grown in Yue territory and cocoa will in time become a major export of the Champa Empire, while it will take centuries longer for a pharmacist in the service of the Northern Wei (Toba) emperors to discover the use of the decorative tobacco plant (whose leaves some tribes in the uncivilised south chew) as a source of a powerful, undetectable poison.
This is as early as I can think ofr right now, without positing some major changes. Ovviously, the Irish, the Vikings, the Central Africans and the Japanese could also act as middlemen at a later date.