You do have to give them a P45 and they can work out your pay from your tax code.
They also don't know if your tax band is due to salary, loans, deduction, pensions, second incomes etc.
I changed job last year and wasn't asked for a P45 by my new employer. Despite starting near month end, my first month's payslip already had the right tax code.
The days of being stuck paying emergency tax for months finally seem to be at an end!