There was a movie that came out several years ago about an Indian
nanny helping a British woman in India.  I thought it had Natasha
Richardson in it.  The nanny ends up trying to take this woman's
children from her.  What is the name of this movie?

Hi there,

I believe the film you describe is Cotton Mary (1999), a
Merchant-Ivory production starring Greta Scacchi. User comments from
Internet Movie Database:

"A sensitive look at the difficulties faced by a woman in colonial
India, during the period when nationalism was starting to set in. The
story opens with an India-born Englishwoman who goes into labour, is
taken to the hospital, and gives birth to a sickly child. When it
turns out that her milk doesn't come in, a nurse with mixed
British/Indian heritage takes pity on her, finds a wet nurse, moves
into her house, and begins to manipulate the situation to her own
advantage."

"...would have you believe that Cotton Mary could take a baby, ship it
across the river to her sister's compound to nurse, ship it back - and
still have time for her various plots? "
http://us.imdb.com/CommentsShow?159373

Search keywords used:

indian nanny english film
://www.google.com/search?num=30&hl=en&lr=&ie=ISO-8859-1&newwindow=1&q=indian+nanny+english+film

I pretty sure this is the movie, based on your plot description. Greta
Scacchi and Natasha Richardson are quite similar in looks and age.
Please ask for a clarification if this is not the film.

Best wishes,
robertskelton-ga

Thank you so much!  This is definitely the movie I was looking for. 
I'm going to rent it at the first chance I get.

I am not certain this is the movie you're looking for, but I'll toss
it into the air and see where it lands.

"My Daughter's Keeper," also known as "Au Pair," has a plot somewhat
similar to what you describe. However, it takes place in Africa rather
than in India. The mother in "Au Pair" is played by Jocelyn Broderick.
I've tried to find an online photo of her, and have not yet located
one, so I can't say whether she resembles Natasha Richardson. Since I
have not seen this movie, I have no further details. Perhaps another
researcher will take up the challenge.

Here is a brief synopsis of "My Daughter's Keeper" with a photo that
might possibly help:

http://www.ifilm.com/ifilm/product/film_info/0,3699,2346244,00.html

Pink, might try her sister(?)Jolie/Joley Richardson.