(scroll down to the section on Marriage in Medieval Europe.)
There won't be any major changes in how people can get married, or when- it'll still be between two consenting adults who are not already married who are not too consanguinous- mind you, that's already shifted as far as the church is concerned. These days, the CofE is quite happy to officiate at the marriage of first cousins, and of those who are already related by marriage (e.g. Price Harry would be allowed to marry Pippa Middleton, to give a real-life example).
The church may claim that their marriages are sanctified by their God, which is fine- but not everyone believes in that God and not everyone has had a marriage sanctified by that God. I know that the C of E is the state religion, but marriage is already not defined by the church's ceremony. All the married Jews, Hindus, Muslims, Atheists, etc out there- their marriages were not sanctified by the Christian God. Anyone want a nice, paranoid slippery slope argument? What might be the long-term impact of establishing as a precedent the idea that the church gets the final say in who is and isn't married? If the only way someone could legally call themself married is if they has a full church ceremony? Serious religious discrimination right there- not just against non-Christians but also, potentially, against various non C of E denominations.
Last thing- there always have been ways to end marriages. Even when the Church banned divorce it was still possible to get out of a marriage via annulment. One could generally find some reason why the marriage wasn't valid in the first place. Henry VIII's main argument was along the lines of "She had been married to my brother, so I shouldn't have married her- the Papal Dispensation to permit the marriage clearly wasn't sufficient to make the union valid in God's eyes- behold the lack of children!"