Hi there "allthebuttons", and Welcome!
I hired a soundproofing company...
Something doesn't feel right here....
Let's start with the basics: What are you trying to achieve? What was your brief to this company? In terms of isolation, I mean. (Most acousticians don't use the term "soundproofing", because it's an oxymoron: it is impossible to "proof" anything against sound... but anyway... ) You must have explained to them what the problem is that you are experiencing, in terms of sound levels inside the room and outside, and you must have told them what your goal is regarding those sound levels. Please tell us the same thing, so we can figure out if this "soundproofing company" is going about it in sound manner. (pun intended!

)
the two side panels were the wrong width, so currently only the central panel is installed.
There does not seem to be any framing in there! I can see where the center unit ends, and the empty space where the side unit will go, but there is supposed to be a framing member of some type in there, to which both units will be attached and sealed.... I don't see that. Are they going to put that in when the new side panel arrives? Or are they not going to do anything there, just leave the units touching each other but not attached to a structural member? Red flag...
The glazing has been fitted directly onto the plasterboard ceiling (see picture)
It's hard to be sure from the photo, but it looks like there's a gap up there! Between the window unit and the ceiling, there seems to be a gap: the unit is not tight up against the ceiling? Is that the case? Maybe you can try to check, using a mirror and a torch, to see if there's gap there. If so, that's a major problem. In order to "soundproof" (="isolate") anything, there can be no air gaps. Zero. Zilch. Nada. Nothing. The reasons is very simple: if air can get through, then so can sound. Sound is just vibrations moving trough the air, so if there is air connecting one side of a barrier to the other side, then the sound will get through. One of the most basic rules of isolating a room, is to seal everything. Then seal it again. And one more time, just for luck! Sealing is critically important. If there really is a gap up there, then that's a red flag for sure. It can be fixed, using backer rod and suitable caulk (flexible: never hardens), but that should have been done when they put the window in.
which means the plasterer will have to work around the secondary glazing in-situ.
I would not plaster up to the window unit: I would stop a bit short, then put some type of trim up there, to hide the joint. Maybe quarter-round? Or some type of cornice? There's some pretty decent looking cornices made from various types of foam, these days. You could try one of those.
That said, I'm very surprised that they did not finish the ceiling up to the ORIGINAL window frame! It's not painted up there! There's no trim! That will be forever visible, and ugly....
Is this normal? Will it help with soundproofing?
Only if the rest of the room is also "soundproofed"! Think of it this way: In many ways, sound acts like water. If you wanted to build an aquarium, you would need to put glass on all four sides and the bottom, to keep the water in. If you take out the glass on one side, the water comes gushing out all over. And if you used something far less "waterproof" than glass, such as carpet, sponge, cardboard, etc. the water might not gush out, but it sure would leak out! And it would not matter that you had excellent glass on all the other sides: Water would still run out of your aquarium. The same applies to your room and sound: All of the room needs to be isolated to the same level. Putting extra glass in only one side of your room, is sort of like taking an aquarium where all of the sides are made of carpet, and putting glass in just one side: it still leaks water. Your room will still "leak" sound.
In simple terms, sound will take the simplest path out of the room, the path that offers least resistance. Just like water.
So, if it really is the case that the "weakest path" in your room was the bay window, then doubling-up on that would indeed be a good thing. But if the rest of the room is only as good as the original bay window was, then there won't be much improvement. For example, if the ceiling happens to be thin and light weight, then that would be a weak path that should have been "beefed up" before the windows went in. If the door happens to have gaps around the edges, then those should have been sealed, as that would be a weak path. If there are electrical boxes in the walls for plugs, switches, and light fittings, then those should also have been "beefed" up with more mass and air-tight seals. If the floor is lightweight and has gaps in it, or around it, then that should have been dealt with too. If there is a fireplace in the room, that should have been removed and bricked up. If there is a ventilation system (HVAC) with registers in the room, they should have had silencer boxes put on them. Etc.
In other words, if you want to "soundproof" your room, you need to soundproof the ENTIRE room, all to the same level, not just one part of it.
It may well be that the bay window was the weak point, in which case you will be getting a noticeable improvement... provided that the new windows are installed correctly! (With framing... fully sealed, air-tight).
Let's hope that that is the case, but from looking at your photo, I'm not convinced...
Please proved more details of the room: photos of the rest of it, description, measurements, what the problem was, and what your goal is.
For now, color me skeptical....
- Stuart -