My main question is - do stairs have to be built into a supporting
wall?

That will be for the structural engineer to decide. There are all sorts of
possibilities, not all requiring structural walls. Also, the stud partition
walls might be structural themselves anyway.

If there really is a dearth of structural walls in the house, this will be
more problematic in terms of strengthening the loft floor structure than in
placing stairways. If you have spans of greater than 4m, it could get
tricky. You might even need to replace some strategic stud walls with
concrete block. Alternatively, it may be possible to install some large
steel beams. You really need a decent structural engineer to decide.

Christian.