Sorry, but the QI boffins were (as is unfortunately often the case) completely wrong. Waldseemueller even annotated his original map, clearly stating he used the name "America" (or "Terra Americi") in honour of Amerigo Vespucci (or Americus Vespucius, in its latinized version, as was customary in those days -- and it was just as customary to use the Latin first name in those days).

We also know that Mercator based his map of the new world partly on that of Waldseemueller and it is safe to assume that neither had even heard of "that Welsh dude". Besides, we know Vespucci went to the Americas: he published a bestseller about his voyages and was arguably better known than Columbus at the time. Of Amerik(e) we only know that he was supposed to have funded Cabot's voyage to North America; in fact, just about the only thing we know about him, is that rumour that America might have been named after his surname. No books, no atlases, no fame...

It's not because Stephen Fry claims something, that it is also right...