So, assuming a semi with a party wall between you and theirs going full width of the garden up to or astride the boundary with the new work.
So, yes. If the wall between your shed and theirs is shared, they need to serve PW notices. If they are building up to the boundary with the new structure or adding a new wall across / on the boundary where no wall exists, they need to engage with the PW Act. Also if they are raising the shared shed wall or are putting in footings below your foundations.
Almost certainly if you have a brick shed it will be on a raft foundation and their new extension footings will be deeper.
I wouldn't go in too hard at first but take some photos of the area, the back of the house, the shed, download the planning to find a local surveyor to discuss things (e.g. www.partywalls.org.uk/find_a_party_wall_professional ). A lot of one-man bands can be reasonably priced (but not all!!).
Ask the neighbour what they plan to do about the PW Act side of things.
If they get their own surveyor, you can sound them out and see if you want to use the same surveyor to keep their costs down. For this kind of thing just agreeing to the work is an option but you're in the hands of the gods as to whether they get decent builders. Not ideal unless there are high levels of trust. There's scope for people tramping over your garden that you might want a degree of control over.
Even if you love your new neighbours, ask they get a schedule of condition done anyway and in the meantime blitz the house with photos yourself, just in case they kick off early.
Most surveyors should be able to give some free advice.