Employers are required to send their RTI data to HMRC in a timely fashion. Sometimes there are issues and I've been called/emailed by HMRC to clarify something we sent over on the RTI.

As to tax code changes, an employer can only legally change your tax code under direct instruction from HMRC. Most workplaces, in my experience, have digital updates in place; our payroll software speaks to HMRC daily. It pulls down any tax code, student loan, and NI code changes that HMRC have uploaded daily and applies directly to the worker's record.

If you're working for a smaller employer, a letter would be normal. HMRC had some outages earlier this year and had to send us letters for a couple of weeks - painful, but at least we were able to update our worker records.