Autoclaves definitely come under the Pressure Systems Safety Regulations 2000.   Ours do. Look at the definition of a pressure system and in particular of a “relevant fluid” – which in this case means steam at whatever pressure.

I would expect an independent test and inspection myself. Autoclaves are potentially very dangerous, and you need more than a visual examination. We had all our pressure systems (and in my department I had more than 200 items) inspected by engineers from the engineering arm of an insurance company. It was not unusual for an autoclave to fail and need attention before the certificate was issued. Autoclaves always had a check at pressure to see that the relife valve actually lifted at the correct pressure.

I'm going to make a bold statement and suggest that the greater majority of employers who have simple pressure vessels in the workplace will not have an IEng qualified member of staff. They will more usually have access to suitable competency via their Insurers?

The type of autoclave I'm dealing with doesn't enable testing of the PRV (or it might be a bursting disc) function.

Just as an aside - some years back I did a lot of work with Dentists and Vetinary surgeries that have usually multiple autoclaves that are in almost constant use.  The company I worked for sought clarification on the application of the regs to such autoclaves and discovered that a generic written scheme from the manufacturer was acceptable but that they still needed examination and testing by a competent person (insurance engineer?) as per the regulations.   Incidentally at that time I saw many cheap autoclaves that were imported from  Germany (made in various Asians countries but marketed by a German supplier) that didn't complay with the regulations at all and could be opened while still under full temperature and pressure.  I witnessed this happen in a vetinary surgery and the vetinary nurse involved suffered severe scalds to her hands, arms and face.  Moral - check the manual very carefully.