First, create a tiddler called WikipediaMetadata and set its url field to https://en.wikipedia.org/wiki/. (You could also make a tiddler containing only the URL, in the text field, and create other tiddlers if you needed to maintain other information about Wikipedia; it's up to you what makes the most sense.)

Then in WikipediaLinkTemplate:

<a class="tc-tiddlylink-external" href={{{ [{WikipediaMetadata!!url}addsuffix<articleName>] }}}><<articleName>></a>

And to use this tiddler:

<$set name="articleName" value="Aardvark">
  {{WikipediaLinkTemplate}}
</$set>

Notice that treating the WikipediaLinkTemplate as a template by placing || prior to its name in the transclusion is not necessary, as it does not access any of the fields of the current tiddler.