Why shouldn't the UK have women bishops? Just last week in the UK, the Church of England voted on the concept of having women bishops (who would then be eligible for appointment to the House of Lords, but that's not relevant.)

They voted no.

On the morning TV show, "The Big Questions", last Sunday, one of the pundits in favour of that decision stated that the church had "scriptural authority" to back the decision up.  This went unchallenged, and so was not expanded upon.

What is the scriptural authority behind this decision?
(Full disclosure, I am an Anglican / Episcopalian. I am 100% behind a very, very strong view of Scriptural authority, though I believe in the case of female priests, the old interpretation is not really in line with what Paul or God wrote. As such, I think it's totally okay to have a female priest- but for Scriptural reasons. )

1. There is a scriptural case for not allowing women any positions of authority in the church

In 1 Corinthians, Paul says that he does not permit a woman to speak in church. Additionally, completarian theology reads several passages in Paul's letters to be clear that a woman is not to exercise "headship" over a man. The Scriptural case is laid out pretty well in this answer, which basically says that there are many Scriptures that say "Women shouldn't preach," but that aren't as clear cut as one might assume.

2. If a Scriptural case exists, then it is a de-facto ground of support

Whether one agrees with the interpretation or not, there is thus an explicit argument in Scripture that women are not to have priestly authority. In the same way that the Queen of England is bound to certain "constitutional" limits, proscribed by tradition, so to the Church is bound to traditional arguments, including certain interpretations of Scripture. 

The central debate amongst Anglicans today is the very question of the role of Scripture.  "Progressives" in the Anglican Tradition believe that Scripture is a source of theology, but not the only one. Traditionalists, including the vast majority of Africans, who outnumber English Anglicans several fold, disagree. As a church of compromise (Anglicans promote the via media - the "Middle Way"), the role of Scripture is the crux of the question- not women themselves. Traditionalists can rightly call on Scripture by looking to Paul, to Judges, and to other references. (Again, will expand when not on a phone)

3. Pragmatically, the question of women priests has been settled, but bishops are an open case

Because society has changed so much, universal acceptance of denying all female roles is no longer an option.  Many, many congregations views denying women a platform to preach as a societal anachronism.  (Again, full disclosure, when I look at the relative position the early church gave women, see, for example 2 John, which was addressed to the elect lady and her church, I am convinced that the Gospel was revolutionary in the freedom it afforded women when compared against the original society.)  As such, as the relative status of women has changed so dramatically, many congregations actually desire both male and female teaching and oversight.  Again, the Anglican via media has a straight-forward solution - let individual congregations decide.

The issue with bishops is that their purview is simply that much larger.  There are thousands of churches, but only hundreds of dioceses.  Additionally, parishes in the Anglican tradition do not have set boundaries, whereas dioceses do* (yes, there are execeptions, but by and large this is true.)  

As such, accidents of geography necessarily override individual congregational preferences.    In the same way that both Republicans and Democrats in the U.S. are forced into the same Congressional seat, so too, both pro-women and anti-women adherents are forced into the same district.

In other words, the compromise doesn't scale to the diocese level.

Because the pragmatic solution doesn't scale to the diocesean level, debate exists there. Anglicans haven't settled this issue, and so, until we do, there will be decisions that each faction vociferously hates.  

4. The bigger issue is Scripture

Ultimately, even the deciding factor - meaning the role of Scripture, how it is interpreted, and how it is applied in this context, will have unintended precedent setting effects.  

If, for example, a female bishop is forced on a congregation that reads Scripture to exclude them, how will they feel when a homosexual bishop is chosen?  The United States has, at the time of writing, two.  Again, remember that this is absolute heresy in the majority of the Anglican world (Uganda, Nigeria, etc...).  Their reading of Scripture would be very simple - 1 Corinthians 5 - expel the immoral brother!