Well, this thread was originally specifically about the Belgian response to the full veil, so Muslims and the full veil are being debated.
The association of the full veil and the burqa with fundamentalism is the case in Wahhabi-dominated Saudi Arabia, which spends billions of pounds sponsoring schools, educating clerics, building mosques, etc., in the west every year. Wahhabism started about 200 years ago, founded by Mohammed ibn Abd al Wahhab, and based on earlier reform movements. It is characterised by severe simplicity of worship of God, among other aspects (unitary nature of God, no revering of any other people, or things, or saints, to put it very simply and possibly therefore to misstate -- apologies in advance). Wahhabis consider Wahhabism to be the only true form of Islam. It is intolerant of other forms of Islam, especially Shia Islam, and seeks to make over the Islamic in its own image. It sees the world in terms of Wahhabism vs. infidels. They are very open about this. It's not a secret.
The purpose of the Wahhabi-dominated Muslim community is to become the living embodiment of God's laws. There are various rules about the correct practice of religion and what this entails, among which is the insistence on modest clothing for both men and women. However, while men are left to their own devices with a few guidelines (for example, the ban on silk, and gold) women's clothing is subject to specific rules, including the full veil or the burqa, which are seen as the only correct expression of what it means to be a Muslim woman in terms of clothing (there are other requirements besides clothing).
They adhere to a strict and literal interpretation of the Koran. Wahhabis reject reinterpretation of religious issues (which essentially means social and political issues in both the Saudi Arabian context and, importantly, in the context of the wider world) that are considered already settled by earlier jurists; here they are in conflict with reform movements within Islam that took place in the 19th and 20th centuries that sought to reinterpret parts of the Koran to bring Islam more in step with standards prevalent in the west, notably in areas such as gender relations, family law and participatory democracy.
Full veil therefore = religious fundamentalism.
I do see it kicking off as more and more women adopt the full veil or the burqa. I think it may end up being skirted by treating the apparel of women as a security/ facial recognition issue, but fundamentalist mosques and firebrand Imams are already seen as a security problem in Britain, and since facial recognition is a security tool, maybe there's some justification for lumping the veil issue in under the heading of security. But the deeper question is British culture, as the full veil is another side of the fundamentalist coin.