"I need to stay on good terms because the builder said scaffolding will need to overhang over their property for the extension, so I don't want it to seem I am threatening them with costs." the lean to does make this a party wall, therefore whether your neighbours like it or not, you can erect the required scaffolding if you follow the party wall framework (this doesn't involve the appointment of a party wall surveyor).
"so I don't want it to seem I am threatening them with costs." why is this threatening them with costs, you need to form some boundary on the relationship with your neighbours as well as the physical ones.
"What can be done about their lean to? They've built it right up against the wall, so shouldn't they have notified me or something?"- Yes,
You need to ask them if they have any documentation, agreement related to the wall, do they have a party wall reward for it? If not then it is trespass.
You need to discuss these issues with your neighbour. The situation is that you could demolish their wall for your works given their is no agreement for it. Off course you won't do that, but they will need to pay for the reinstatement of their own structure, or as above Alternatively, you can offer them an alternative, if unnecessary party wall surveyors can be avoided, you would be willing to ensure that your builder takes care of the necessary steps. If not then make sure they paid for the work you need to do on your side because of their trespass.