I am pretty sure that your tax code has to be written somewhere on every pay slip. It might be hidden within a deduction line item instead of being explicitly noted.
e.g.
Income Tax (1060L) - £XXXX.XX
National Insurance - £XXX.XX
Student Loan - £XXX.XX
instead of
Tax code: 1060L
Also, from what I understand, only HMRC can change a person's tax code, so if you have been put on the wrong tax code, phone HMRC and ask for it to be adjusted. They'll then send the updated tax code to your employer on your behalf.