Was there ever a period where uni students were exempt from paying income tax and national insurance?
Grants and most bursaries are not taxable. In the olden days, you could live on the maintenance grant, so tax never came into your head. Everything else is and always was taxed without regard to your student status.
Income from employment was always subject to tax. However, there was an exception that allowed you to be paid without tax deducted at source in certain circumstances, where you did not expect your total annual income to exceed the personal allowance.
This might apply if you had a holiday job at a decent salary (so liable for tax if you're there for a year, so you end up with PAYE deductions), but you were only going to be working for, say, 10 weeks (so the total was under the tax threshold).
If you could be arsed filling out the relevant form, and your employer's payroll team was competent enough to implement it, they were allowed to pay gross of tax, though still net of NI contributions.
(Here we go - form P38(s), withdrawn in 2013. https://www.gov.uk/hmrc-internal-manuals/paye-manual/paye46045)