According to its wikipedia article, the Anglican publication 'Common Worship: Services and Prayers for the Church of England' was launched in the year 2000 and was a successor to the Alternative Service Book (ASB) of 1980, which was an authorized alternative to the Book of Common Prayer of 1662.
The form of the wedding service in the first trial use volume of 1965 (of the ASB) retained the question by the Priest, "who giveth this woman &c." as did the 1977 publication of the marriage service. However, in 1977, The word "may" was added to the rubric (the instructions to the Priest printed in the book), so that while formerly it was obligatory to ask, in both options in A and B in the 1977 volume (and presumably the ASB in 1980, the asking "who giveth..." was at the discretion of the Priest.
It should be noted that although each of the independent Churches in the Anglican Communion are autonomous, and many (if not all) prepare their own version of the Book of Common Prayer. Tthe development of the various Books of Common Prayer of the various constituent bodies of the Anglican Communion does not take place in a vacuum, however. In the 1960's and 1970's, the development of the Book of Common Prayer of the US Episcopal Church led to the revision of 1979, in the marriage service of which there is a note after the declaration of consent (on p. 425) which provides that "If there is to be a presentation of a giving in marriage it takes place at this time. See p. 437. The additional directions there provide two options: that the woman may be given(presented) in marriage to the man, or that the woman and man may be presented to be married to each other.
While I have no direct knowledge that those who prepared the ASB were familiar with the US BCP, I would be astonished if they did not.
The reason for these changes, and others like them, was to adapt the forms of service to be an appropriate alternative (to the BCP) within a 20th (and later, 21st) century context (the BCP had not been revised for over 300 years, efforts to do so had been defeated 40 years earlier - source). In particular regard to the wedding service, the revision process accommodated a more modern understanding of life and marriage that would separate the cultural norms of patriarchal societies from eternal scriptural principles, particularly cognizant that: a daughter is not the property of her father until she is given over to become the property of her husband. A judgment was made, that while the reverse of the latter statement is not necessarily anti-thetical to biblical principle, it is not necessitated by it either.
Another way of summarising this is to say that the Anglican Communion (in general and the Church of England in particular), has been subject to the influence of Feminism. Saying this is not to imply that all aspects of Feminism have been incorporated into Anglican polity - where this influence has been generally recognised and adopted, it is usually termed 'egalitarianism'.