I seem to recall somthing similar from my visit to the UN in New York a few years back (leisure not business).

The following webpage states so too:

https://www.un.org/en/oaj/unjs/why.shtml

Although you would obviously need to do your own research to verify this and apply it to context.

As far as I know the UN has limited immunity in some circumstances not blanket immunity.
In which part of the world are you asking about?
Diplomatic immunity is not a “get out of jail free card”, which is how it is portrayed in some TV shows. It exists is as the result an agreement between states or between states and some international bodies like the UN, to waive the right to bring legal actions against individuals working for those states who have been invited to set up diplomatic missions in host country, if such actions were to prevent them doing the job they had come to do in the host country.
I am not sure how someone could claim diplomatic immunity in the case of Health and Safety law, certainly not in the case of the UK.