I have just read heidi to my 4 year old and we didn't get bogged down with the (imo loose,) religious context. She just loved the concept of a little girl living in the mountains. It is her favourite story.
Other favourites I've just thought of are "Flat Stanley" this is a series of several different stories all about stanley, they are very funny, also "The selfish giant" and of course all the other short stories by Oscar Wilde esp "The happy Prince". My daughter also loves to listen to the original Hans Christian Anderson stories. We have a big compendium which she takes with her everywhere. There aren't many pictures in our version and when there are some, they are tiny line drawings, but she just likes to lie in bed listening to them. She is fascinated by old language.
Abridged texts could be good too. The usbourne young reader series is fantastic and contains titles such as Robinso Crusoe, Treasure Island, Swiss Family Robinson to name a few. My children loved these when they were very young and then were not afriad to progress to the original texts when competent older readers at about 9 or 10.
Oh, there are so many books, i wish we could just sit down all day long and read them all.....!