I would go back much further than Blair. The roots of that situation can be found ultimately in the British empire, which lead to the creation of Pakistan and Bangladesh, and immigration from those countries into Britain (notably from Sylhet in the 1970s).
Crucially, South Asian people are endogamous, meaning that they generally prefer to marry within their communities and moreover within the family: cousin marriage is common. So there has been consistent secondary immigration from the subcontinent as South Asians marry spouses from their communities from their countries of origin and bring them to the UK.
The average Bangladeshi or Pakistani citizen's religious views are about as welcome in Britain as the average Briton's secular views would be welcome in Bangladesh. Which is to say that they are disjoint and cannot be reconciled much at all.
Crucially, South Asian people are endogamous, meaning that they generally prefer to marry within their communities and moreover within the family: cousin marriage is common. So there has been consistent secondary immigration from the subcontinent as South Asians marry spouses from their communities from their countries of origin and bring them to the UK.
The average Bangladeshi or Pakistani citizen's religious views are about as welcome in Britain as the average Briton's secular views would be welcome in Bangladesh. Which is to say that they are disjoint and cannot be reconciled much at all.