I'm not sure about the situation in England, however not too long ago a similar situation happened in The Netherlands. There was a big discussion, so most sources are in Dutch, mainly the Dutch Wikipedia with it's references.
There are two groups who do not vaccinate their children:
A) People in the The Anthroposophical Society believe that vaccinations affect the immune system.
B) Orthodox protestant reformed groups, which is a very traditional church with it's own clothing style, schools, hospitals, politic party etc. They do not watch TV or use a computer because the bible does not say anything about modern technology.
The main reason they do not vaccinate their children is because of Sunday 10, Q&A 27 from the Heidelberg Catechism, which reads:
Question 27. What dost thou mean by the providence of God?
Answer: The almighty and everywhere present power of God; whereby, as
it were by his hand, he upholds and governs heaven, earth, and all
creatures; so that herbs and grass, rain and drought, fruitful and
barren years, meat and drink, health and sickness, riches and poverty,
yea, and all things come, not by chance, but be his fatherly hand.
They believe that all good comes from God, and sickness is a punishment from God. They believe that God wants their children to be healthy He'll make them better. If God wants them to become disabled due to an illness, it's His will, and they accept that.
There is much discussion about this topic within this denomination. Some of their church leaders say that it is something between the you and God to decide what is right. If you feel that the vaccination places technology between you your dependence on God, you should not use vaccinations. However if you do not feel guilty when using vaccinations you are free to use them.