Royal Greenwich spans an unusually wide range for one borough: a World Heritage Site at its historic core, Victorian terraces through Charlton and East Greenwich, inter-war suburbia in Eltham, and new-build riverside development along the peninsula. The right answer varies enormously across those.
Of the borough's 23 conservation areas, six have had permitted development rights withdrawn for external alterations: Ashburnham Triangle, East Greenwich (two separate directions), the Progress Estate, Rectory Field, Shrewsbury Park Estate and West Greenwich (also two directions).
The Progress Estate deserves specific mention: it's an Arts and Crafts-influenced planned estate of considerable architectural coherence, and the direction there exists precisely to prevent piecemeal external change. Expect scrutiny of anything roof-mounted.
Greenwich also has a borough-wide HMO Article 4 direction, which does not affect solar or heat pumps.
Maritime Greenwich is a UNESCO World Heritage Site, and its buffer zone extends well beyond the obvious landmarks. The protected sightline from Greenwich Park towards St Paul's is a genuine planning consideration, not a formality. If your property sits within the buffer or on that sightline, roof-mounted equipment gets assessed on visual impact in a way it wouldn't three streets away — and that assessment can apply even outside a designated conservation area.
Eltham, Plumstead and much of Charlton are where the straightforward projects are: inter-war and post-war semis and detached houses with side access, proper gardens and largely unconstrained roofs. Cavity wall insulation is often the highest-return first move on this stock.
Central and west Greenwich is the opposite — Georgian and early Victorian terraces, tight plots, heavy heritage oversight. The peninsula's newer riverside blocks are almost entirely flats, which means no permitted development rights and freeholder consent required regardless of what the planning map says.
For costs, grants and how the national rules work, see the heat pump guide, solar guide, air conditioning guide and insulation guide.