In nursery, they play games such as I Spy, and are exposed to lots books and stories and nursery rhymes, which is very important. At Reception, they play more letter games, and learn that letters make words, and they learn rhyming words. From January, in Nursery, they will start on phonics, with songs and signs (using the jolly phonics method at our school). Then they will learn how to blend the sounds together and sound out three-letter words (mum, dad, dog, cat, etc). At the end of Reception, they should be able to sound out more words on their own, and blend them. And from year one, they start reading short simple sentences. Then they learn more complex words and are encouraged to read every day, and learn to write by forming the letters correctly. From then on they progress very quickly in my experience. In no time they are able to read books with some help.
In my eldest son's class, there were three or four children who could read a bit before starting reception, and one of them could read very confidently at the age of five. My eldest learned most of his phonics at school and with support at home, whereas my youngest can already read simple books and he is 4.5. He listed and joined in when DS1 was learning at home and he picked it up very quickly.
In my experience, it's not true that teachers prefer that children can't read before starting big school. DS2 can read (in reception) and the teacher is very supportive in giving him books and supporting him in class. I think they are keen that parents use the phonics method, as it is generally the method used in schools.
I think that teaching children to read before they start school is up to the parents and there is no right or wrong. But now, DS1 can read as well as the little girl who could read before starting school. They do catch up, especially if they are interested in books and keen on reading.