I was briefly researching this because I wanted to know what the mandatory procedure was, but I didn't get to the bottom of it.
The whole gist of that article is that you should, (but it may not be legally required). For instance, the article also says:
"To answer the question: Yes, a Category S car, which has sustained structural damage, does require a new MOT test after it has been repaired and before it can be driven again. This is to ensure that the vehicle is roadworthy and safe to operate on public roads following the structural repairs."
There was also something called the Vehicle Identity Check by the DVLA, which applied to category S cars, but ended in 2015.
A guide about the VIC scheme.
www.gov.uk
So, I remain unsure what the mandatory requirements regarding vehicle inspection after a Category S write off are. I would have thought there needed to be a proper engineering inspection by some sort of government approved scheme, but I couldn't find anything.