Calling Nigel sounds like a good idea. On most aftermarket wheels I have seen the horn push is sized to fit the center hole of the wheel. On the Triumphs I have worked on the OEM horn push has been sized to fit the center of the hub instead.
Some aftermarket wheels have horn pushes that come with wires. The Spitfires horn pushes I have seen use a simple metal tang on the back of the button. The spring loaded brush pushes against that tang at the top and a slip ring on the steering column below the hub.
This may not be an option for your particular wheel. However, when I fit an aftermarket wheel to our Spitfire, rather than buy a new, longer horn brush I made a little extension piece out of brass. With the spring loaded portion of the brush pointed down against the slip ring I placed the extension cap on the top end of the brush so it would push against the tang on the horn button. Since the top end of the horn brush does not move relative to the hub or wheel, the simple cap is securely held in place and I didn't have to buy a new part.