How will you know which tax code to put me on?

When you join a new Health Board (or any organisation in the UK), the organisation is required to assign you a tax code on joining.  

If you have received your P45 from your previous employer – and passed it on to your new employing board within the payroll deadlines - then your new health board will use the code on the P45 which will be the same as the code used at your previous employer.

If you have not yet received your P45 from your previous employer then your new employing board, by legislation, to assign you a temporary tax code, based on your salary.  That is either BR or 0T, dependent on your earnings in that role.