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Double doors lead to dining room. There is a wall between the dining room and "utility" that apparently used to be an arch. That will likely get fucked through pretty quickly (some of the plaster board has already been caved in by boisterous teenagers, so we know it isn't load bearing). Lean to is just garden tool storage on the outside, there is a separate cupboard on the inside that houses the boiler and some other crap.
https://www.yopa.co.uk/properties/details/191242
If we do extend, we'd want big fuck off doors and roof lights for just that reason. But that might be a 4 or 5 year away thing.
What room is behind the double doors? Obviously you will lose light in there, so I would be considering a nice big roof light above if you extended across. Anything important in the little lean to? gas meter etc?