Anime is too varied to tell whether something has really been influenced by it or not… Things like Teen Titans and Totally Spies are based on around what some people think that anime is, but inescapably American in spirit, in their story structures and character stereotypes.
I could however point you towards to some 'mature', adult-aimed animation which originated in the UK (to varying degrees):
Watership Down
The Thief and the Cobbler
The Secret Adventures of Tom Thumb (released here by Manga!)
Y Mabinogi
Though all of those have been quite forcibly and wrongly aimed at children, just because it's only possible way to make a feature animation here. But in terms of short films, the UK is really something - there's Phil Mulloy's Possession, Suzie Templeton's Peter & the Wolf, Philip Hunt's Ah Pook Is Here, Marc Craste's Pica series, the Brothers Quay and Aardman and many more… We may suck when it comes to cel animation (that's the forte of the Japanese and Canadian) but British, like Czech and Russian stop-motion is famed throughout the world.