Spiral spring.

[penfold]

Need a bit of 'spring' in it, piano wire or similar is needed.

 

[Graham_Wright]

Why not place 2 No 10mm discs on a shaft, One disc with a thin 2mm shoulder to keep the 2 mm gap. The shaft made from a bolt. The 2 discs held by a nut wound up tight. . A longitudinal slotted hole in the shaft & shoulder to start off the wire. The head of the bolt held in a 3 jaw chuck. The other end in the tailstock revolving centre.
Now the problem is getting the wire to fit tight to the coils. Fit a piece of steel in the toolpost like a lathe cutter but with 2 small bolts in the side of it held parallel to the lathe centre. The wire passes between the bolts & is kinked, thus bending it . That makes it pre bent & want to curl tightly into the gap between the 2 plates or discs. It also adds friction to keep it tight. When you reach the full diam of the discs stop. Then undo the nut & separate the discs. The slotted hole allows one to draw the wire along the bolt & cut the wire from the slotted hole with a dremel so it can be released from the bolt.
Now one can pull the middle out & form the cone. Obviously it will not be totally small in the middle but you have not said what it is for
I do not know enough about heat treating to know if it should be treated at this stage or not. Others may comment. Also the type of wire will have a bearing on the work. But i believe that as the OP has done spring winding he will have some idea of wire type best suited to the task.

#9 shows the result of that technique. The wire needs to be wound unto a diameter smaller than its to be relaxed diameter. Winding wire on top of the previous turn does not achieve that. Each turn forms the "arbour" for the following one.
The conical one was wound onto a tapered arbour. The increasing turns do not increase enough which means that the spring will not compress down to flatness.
The answer is a faster taper which will be more of a challenge to wind.
The wire I am using is hard drawn phosphor bronze 2mm in diameter left over from MastaClimba brake springs.
I have been holding the end by hand which is stupid as, when I run out of length and let go, the free end attacks me. I will resort to Molegrips next time.

 

[Graham_Wright]

Need a bit of 'spring' in it, piano wire or similar is needed.

See #3 and #22.
Material is ideal (and piano wire would rust).

 

[Daydream believer]

#9 shows the result of that technique. The wire needs to be wound unto a diameter smaller than its to be relaxed diameter. Winding wire on top of the previous turn does not achieve that. Each turn forms the "arbour" for the following one.

I understand that it will not stay in place, hence the reason for "pre bending " between 2 points as it goes on to the lathe. This will give it some "memory" The coil on the lathe is effectively holding the coils apart, not together. . The wire will not approach from the front of the lathe in the normal way but from the back , over the top so it is bending round one of the the bars as it goes in on the underside of the final coil. The other bar preventing the wire from coming over & smacking you in the face. Its distance from the first governs its radius & amount of "memory". It needs to be close & tension needs to be adjusted so it does not forget the "memory" on the way in
The bottom bar goes at 6 o clock & the top one at 2 o clock approx.
The bottom bar is as close to the lathe centre as the discs will allow to reduce the straight portion which is where the wire gets straightened again if you get it wrong.
The princile is similar to going through a set of rollers & then storing the rolled wire on a coil. It is just that the final coil takes a bit of practice. As you have found

 

[Graham_Wright]

I run the lathe in reverse and preferably with herself on the "go" lever.
I have hopes that the pliers will allow conversion of one of the two samples to serve.