My DH is a structural engineer so I will pass on what I know.
Underpinning in the 1930's will not be of interest to anyone - it is too long ago and no-one really knows what was done. A surveyor should not suggest a slightly out of alignment staircase is anything to do with something that happened 80 years ago! Staircases are internal usually and it woldhave to have been a pretty major problem to move a staircase. What has happened to it in the intervening years might be of interest. Was it built properly?
Subsidence is what happens when moisture is taken out of the ground. Usually by a tree(s) that is too large and the roots are looking for moisture. It can be that a house has very shallow foundations and is not built to withstand changes in moisture levels in the soil. It is more likely there is subsidence on clay soils because they shrink rapidly which means the foundations fail under the ground and the house collapses downwards. Where there is too much moisture in the soil - this damage is called heave. As you can imagine it pushes the walls out of alignment and again is due to insufficient foundations for the conditions. This can also be a clay soil problem and instead of the soil being dessicated, (as in subsidence) it is saturated so it moves upwards, pushing the walls outwards. A leaking drain could cause this, but removal of loads of trees can have the same effect in that too much water is in the soil.
Usually subsidence work is guaranteed for 10 years. If the surveryor says there is historic movement, it means the house is no longer moving, but it has in the past. There may be signs of repaired cracks, inside and out. If a staircase has moved it is likely to have been a very serious case. I will check with DH to see what remedies might have been used inthe 1930's. Today it is very deep concrete foundations, to put it simply. My DH has actually advised that a house be demolished and rebuilt due to heave. You could see from the lounge into the bedroom above because the walls had moved outwards so much. The staircase was no longer adjacent to the wall - it had split away!
A surveyor cannot see underpinning! It is under the ground!!! All you will see from above is repaired cracks. If the house has been re-rendered, you may not even see that! If a house has been underpinned, it is usually a good bet because the problem has been dealt with. Surveyors can be very cautious about cracks and are not qualified to say exactly what is happening or what the remedy should be. As you rightly say that is a structural engineer but you don't want to pay for one if you do not need one.
I do not think you need to mention hearsay of the neighbours. My DH is selling a relative's house that he underpinned 24 years ago. As far as I am aware, the Council did not need to know because it is repair, not a new building. These are insurance jobs normally but building regs could be involved. Repairs are normally specialist solutions tailored to the house and soil. If new drains are involved or there are other building regulation requirements, then the council should know about it. With our relative's house, he gave all the information to the buyer - no-one else did. I could be wrong with more recent subsidence work.
All decent surveyors would notice cracks or repairs to cracks, so having access to a previous survey might help. Subsidence repairs will also have drawings so someone shold have these. However, I would leave it up to the buyer's surveyor. You can only report what you know and it is up to them to check for themselves but usually small cracks in an older house are nothing to worry about. If you can see significant and obvious cracking and windows/doors out of alignment, then that is more concerning. Hope this helps .