While it's not impossible that one of Elizabeth's scholars or one of the Jesuits in Japan might have mentioned to one of the other, I'm not aware of it being recorded. Of note, if it did happen then it must have been through the Jesuits, as they were the ones who involved themselves in Japanese politics and reported back to Europe about it. So it's actually fairly unlikely, given that Jesuits were a Catholic organization at a time of religious strife in Europe (notably when Nobunaga died in 1582, England and Spain-Portugal were at each other's throats, and the Jesuits were deeply intertwined with the Spanish crown).

The East India Company (both English and Dutch) wouldn't be formed until long after Nobunaga's death, and the first official contact between the two nations was probably James I's letter to Tokugawa Ieyasu and his reply, both translated by William Adams in 1613.