If the employee hasn't worked in between the contract / assignment at King's, the tax code will remain the same. The tax code for 2022/23 is 12750L.  If you see a change in your tax code, you should call HMRC tax code line (0300 200 3300).  You will need your national insurance number to identify yourself.  If your tax code is incorrect, HMRC will send the correct code to the payroll provider via a secure portal.

Tax codes: Overview - GOV.UK (www.gov.uk)