Yes, you often see a marriage in a CoE church but baptisms in nonconformist church. The biggest push behind the introduction of civil registration came from the nonconformist churches - as the number of nonconformists grew and particularly the number of well-off, well-educated nonconformists who didn't like having to switch churches on their wedding day! (And for whom the issues surrounding legal illegitimacy and inheritence were much more important).
The law was originally introduced to cut down on 'fleet marriages' and the like - there were special dispensations for those of Jewish or Quaker faith, but everyone else had to marry CoE to be recognised. There was a very sad case in The Times I saw once - a young lady who had married, had a child, and been widowed young; his will had specified his estate go to her, including any inheritance, but when his father died afterwards, leaving money to his son or 'legal heirs', the family successfully argued that neither she nor her child counted towards this.
This was because they had not been married CoE, and despite having witnesses to prove the marriage - including the minister! - by law she was an unmarried mother and her child illegitimate, and although the judge indicated his sympathy for her situation, he couldn't go against the word of law.
There was no end of opposition to the introduction of civil registration, actually, because of this - because it allowed not just nonconformists to be recognised the same as those who belonged to the Church of England, but allowed registry office marriages without church at all. I transcribed once a long, slightly frothy op-ed in the Times of 1836 which claimed that this would destroy marriage - and that registration of births with the government would stop baptisms taking place, and that England would be overrun by the Unitarians and all manner of European immorality, and so on.
ETA: the law was Lord Hardwick's Marriage Act, 1754.
