BI & further education
After they have achieved the necessary educational standard and gained control of their BI, young adults wishing to stay in education (academic or vocational) can use the part of their BI that is earmarked for education to contribute to the ongoing costs.
Parents/carers may require a statutorily-defined contribution (£1,000?) from the remainder of the young adult's BI for rent and board once the child has control of their BI. Reciprocally, they must continue to offer bed and board to the young adult if the young adult is willing to pay this amount, until they reach the age of 18. All children will therefore be able to stay in education until 18 if they wish to, whilst having considerably more freedom to choose not to if they are more interested in the real world than academia at that stage of their lives.
Higher-education funding should work along similar lines. All universities would be independent. The government would declare how many places in each subject it is willing to fund fully in the interests of the nation each year (e.g. with only slight exaggeration and tongue in cheek: tens of thousands of engineers, hundreds of historians, and no MBAs).
The top students (on the basis of their A-level results, assuming that standards are restored and distinguishing between them becomes possible again) upto the declared number in each subject will receive a bursary from the government equal to the average cost of courses in their chosen subject, and can use this to pay the fees at whichever university they go to. If they choose a more expensive university, they can use some of their BI to top up the fees.
Anyone below this standard may still go to university if they can get a place, and can use their BI to assist with costs, but they will have to fund the balance privately (loans, parents, scholarships, part-time work, spending some time in work before going to university, or however). In this way, the top students regardless of background and income will be encouraged and supported to go to university, but the total cost to the taxpayer for higher education will be constrained. Taxi-drivers (and the like) will be paying for fewer middle-class children of ordinary academic ability to spend three years partying and drinking on the ostensible grounds of studying the history of art (or similar).