Ian Stirling wrote:
I have been idly considering, for several years, a lean-to greenhouse,
along the south facing side of the house.
It's nowhere near a road or neighbours, and the house is sunken on
that side, with a 1m path, to a 1-2m wall.

This would be integrated with ventilation, and cooling systems, to aid
in house heating, to reduce bills, and keep it cool.
(as insolation will reach 30Kw or so at peak)

Neglecting this aspect, what is this with regards to building/planning
regulations?
Is it a greenhouse, a conservatory, ...

Does this change if there is a 1cm air gap between this and the house?

It's a temporary structure and you don't need planning permission, although
you can expect a deluge of posts to tell you that you do, by the brigade who
apply for planning permission each time they want to mow their lawn.
Our building control aren't even interested in conservatories