Okay, not trying to be offensive here, but it really baffles me. I like to watch British shows, including gardening shows, and nearly everyone (including the wonderful Monty Don) pronounces it as if it's spelled "anenome"!

And it's not like "aluminum" vs. "aluminium," where it's pronounced differently because it's spelled differently. It's spelled "anemone" over there too, but I've only heard one person (Carol Klein) pronounce it correctly. Even people from other countries who learn British English (e.g., India) seem to pronounce it that way. Why is that??

You learned British English, Ash. How do you say it?