Instead of wire, try copper tubing. At these voltages and frequencies, the wire has a "skin" effect where most of the current travels on the surface anyway. For making perfect-looking round bobbinless coils, I wind them around a cylinder that's smaller than what I want, then they "spring back" to the larger diameter. With this you would probably want to find a cone shape to wind them around.
Won't affect it, this design is a relaxation oscillator that will find its natural resonance.
It often helps to make the tunable primary, though. Wind an extra turn on it, and instead of having that outside lead head back to the supply, use a small battery clamp or something like this -
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- so that you can adjust it for peak resonance with the secondary.