Your employer deducts tax based on the code HMRC issues them via the RTI (Real Time Information) system. Even if you fix your code online, it takes 3 to 5 working days for HMRC to send your employer a new "Coding Notice" (P6 form). Tell your payroll team to expect the update.
If you are a Welsh resident, the financial impact of the "C" prefix is effectively neutral compared to an English resident. Since the Welsh Government has, to date, maintained the same tax rates as England, you are c1251l tax code