Why does Ana speak English so badly? You'd expect her parents to talk English with her at home and young children are very capable of learning two or more languages at once. I did.
It's mentioned at some point in the manga that her parents don't speak English anymore, not even at home. This is probably not likely at all to happen in real life, but it is what it is. So Ana learned some English, but it is a running gag throughout the series that, even though she is not native, she knows more about Japanese customs and legends than do her native-born friends.
Also, think about it. She moved from the US when she was 5. It's been what, 6 years? No way you'd remember a language from that age if your family doesn't speak it at home. I know this from experience, having a first language of Spanish until 3rd grade, when my new family spoke English at home. I'm in college now and don't know a word of Spanish.
Moved from the UK - Ana is English, don't forget.
It's also possible for this to happen with your native tongue, no matter how unlikely it may seem, as what happened with me when I spoke mostly English since primary schooling and now in college, I can barely speak in even conversational Filipino.
Adding some more support to this, I, who am Taiwanese, can assure you that living in the US since kindergarten has made me basically illiterate in Chinese, with only a preschool level of conversational Mandarin and Hokkien/Minnan, despite the fact that his mother still uses both languages at home occasionally.
I, who am Polish, use English (and sometimes also Japanese) for most things (except for taking university notes), and even though I do live in Poland, after just a few years I have to translate what I want to say in Polish because it is natural for me to use English. Not quite the level of "forgetting one's language", but I'm pretty sure I would've forgotten it too if I didn't live in a country where it is used.
Speaking as a Brit who (like Ana) lived abroad as a child (in Norway, China and Japan - 3 years in each country) due to my father's work placements with BP, I can tell you that my parents NEVER stopped speaking our/their native English at home. In fact, English people are known for being bad at picking up other languages, because so many people speak English anyway. I'd suggest it's simply a highly unlikely Plot Device played for laughs.
Nobue's behavior toward Matsuri, Ana and sometimes Miu has raised the eyebrows of some fans. As for Barasui, he said in an omake that he'll draw anything with children in it. As if to drive this point home, he also contributed to the loli series Weekly Dearest My Brother (his submissions were tame but other artists sent in ecchi stuff).
Nobue might not be a Lolicon in its primary meaning ("paedophile"), but Japanese use this word to depict people who like children for other reasons too, and since she finds spending time with them fun, yeah. Besides, trying "old man jokes" on fifth and sixth-graders isn't exactly normal.