You would generally need a meter or so space to build it and for future access, unless your neighbour allows your builder to work in and access via their garden(not likely)
I don’t think ‘permitted’ includes extensions(I’m in Scotland) but I may have misunderstood, ‘permitted’ would apply to changing the roof? I’d email them to get that in writing and you can check on what planning consent you need at the same time.
There are ways of building up to the boundary edge but it’s not popular with neighbours and a big source of falling out, looks like a full planning application to me, which if objected to means another, always, always best to speak to your neighbours about things before they get a letter both shocking them and asking for their permission…it’s almost designed to get an objection.