Wormwood, Wormtounge & Wormtail

As I've been rereading the Harry Potter books, I've been struck by the similarity between the names, or nicknames of at least two of the subject characters: Grima Wormtounge and Peter "Wormtail" Pettegriew. In both cases we have a character who was thought to be a trusted friend or advisor, but was in fact loyal to an enemy instead. Then, in both cases the character in question ended up being called by their nickname by both sides - or even the diminutive "Worm," for the rest of the series.

What I'm wondering is, was there any influence on Rowling from Tolkein.

Also was there more than coincidence in both Tolkien and Lewis having similarly named characters, even though they don't have the same level of similarity.

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