I think our stairs are falling off the wall...

[BabCNesbitt]

…who do I need to speak to?

We have a Victorian terrace house with two rooms on the ground floor and very narrow stairs that go up between the two rooms (so there are walls on both sides of the stairs). For a while after we moved in, 18 months ago, I noticed small cracks appearing along the top of - I think it’s the stair string? The wood at the sides of the stairs that are attached to the wall? - on one side.

Everyone (including a couple of joiners/handymen) just reassured me that it was plaster settling. But the cracks are getting both longer and wider (a couple of mm in places), small chunks of plaster are falling out both next to the stairs and (I think) underneath them, and cracks have started appearing on the other side too.

Do we need to pay for a structural engineer to come out and assess it, or would a reliable joiner be enough? (Not that I’ve had much luck finding one of them, but... )

Reliable joiner should be enough.
Settlement cracks are quite common and if the house had been empty and damp before you moved in it would make it worse. The chances that the stairs are not fixed in and therefore could fall down are very remote. Was a new staircase fitted immediately before you moved in?
If so, and it had been stored in a damp place it could well shrink a few mm. Does any of it move when you walk up and down them?

The house hadn’t been empty, but it had been recently done up by someone who bought it to sell on, and I’m not too sure what kind of state it was in before he bought it. It doesn’t feel wobbly, but there are small nails sticking up just in front of the risers - probably the carpet tack things? (We’ve learned to step around them/wear slippers for now!)

Directly below the stairs are steps down to the basement, so I can see the plasterboard that seems to have been tacked on underneath the stairs and it looks like some bits of plaster have come loose on that.

I would not worry excessively. If the staircase is original then the chances of it falling are extremely rare. I have just removed a 100 year old or more staircase that was FULL of woodworm and it took me most of a day and a lot of effort to get it out.
As it was 'done up to sell' the filler used was probably done in haste and not of a 'semi flexible' type which would be useful on places like stairs where there will be slight movement as you use them.