What kind of certification is sufficient for a conservatory improvement?

TELLIT01

This may be helpful in clarifying whether or not compliance with Building Regs is required.

https://www.lekasystems.co.uk/tiled-roof-conservatory-building-regulations/#post-content

Ena123

So I checked the papers to the house. The 'extension' was built in 1991, one storey. 
The previous home owners, sellers to us, attached a filled form to the document, not able to answer some questions because the structure was there prior to their purchase. It is below 30m^2 area (according to their form they filled).

Nothing was changed from previous owners to those we bought it from on the connection to the 'extension', i.e. those missing doors we discussed earlier. We haven't changed it either during our time here. 

The previous roof was polycarbonate from outside but plastered from underneath. I don't know when that plastering happened. It now has the warm foam sandwich roof with a plastic imitation of slate tiles on top.

According to the latest link to the Leka website above, solid roof is not exempt from the building regulations. Planning permission probably wouldn't apply. 

The glazed area, above the brick wall, could be over 50% of the external wall, it is hard to judge with that low brick wall.
All in all, I have no document confirming this structure, whether extension or conservatory, ever had a final completion certificate. And the current installation company of the windows, doors AND the roof claim that because it was a retrofit of the solid roof (that polycarbonate on top and plastering below, between wooden beams structure), they don't need this certificate either. 

I am still not sure where I stand in this case. 

Ectophile

I am still not sure where I stand in this case.

You have an extension on your house that was never signed off to building regulations.  You could have gotten away with it under the 10 year rule, but you have had the extension modified, and that still hasn't been signed off to building regulations.

As others have said, if you have a door between the house and the extension, then you can call it a conservatory, and conservatories don't have to meet building regulations.