Is that a coil with a single screw in the center of the bottom?
If that is so, you can easily take it apart, take some other screw and put it in a drill chuck and screw the bobbin on that.
I do not have much experience doing this, but it seems logical that the number of turns (especially for DC) is not very critical.
I'll explain:
Let's assume you're grossly out, and put 2 times as many turns on it as you should.
Then the DC resistance will also be approx twice as the old coil. (outer windings are longer).
This results in half the current through the relay.
Each winding will attribute only half to the magnetic field, because of the lower current through the wire.
But you have twice as many windings, so the total magnetic field will still be the same.
You've just built a more efficient relay which needs less power to activate it.
For (old?) reed contacts, you can sometimes find specs for AW = Amps * Windings which is needed to activate it.
This of course only holds true if you use exactly the same wire thickness.
A 230V coil will have very fine wire, which is difficult to handle manually without breaking.
My guess is you'll be better off with replacing the whole relay, unless the steampunk looks are important.
Buying just some loose wire is probably more expensive then a new relay.
Edit:
Oops, with AC it's different.
Twice the number of windings, will result in twice the inductance of the coil.
Whether this is a problem depends on how much of the impedance is from the resistance, and how much is from the inductance.
If you can only source thicker wire, then the voltage for the coil will be less to activte it. (Thicker wire is less Ohms for a meter of wire, and also fewer windings for the same mass of copper). In such a case, fit as many wire as will fit, and use a series resistor (or capacitor for AC) to make up the difference.
For AC it may be good to add a resistor to limit the Q- factor and risk of oscillations.