Do I pay tax on my first £1,000 of delivery income?

No. The trading allowance gives you £1,000 of self-employed income each tax year with no tax and no need to register, as long as that is all you have.

It is measured on income before expenses, so a driver who took £1,400 and spent £600 on fuel has passed it, even though the profit was £800.

Once you pass it you have a choice each year: claim the allowance instead of your expenses, or claim your actual expenses. You cannot do both. For anyone driving regularly, mileage relief is worth far more than £1,000, so the allowance stops being the better option quickly.