After a bunch of debugging and digging into the issue, it turns out this is happening because the TeeChart code is calling GetDeviceCaps with the HORIZSIZE and VERTSIZE parameters, to determine the physical size of the pixels on screen and adjust the circle's bounding rect accordingly. Unfortunately, this call is only valid on a printer, and not on a display device, and it has known issues on Windows 7, which both of the systems in question are using. I've reported the issue to Steema. Hopefully they can get it fixed.
UPDATE: Got a response from Steema, in which they acknowledged the problem and provided a workaround. Copying it here in case anyone else runs into the problem:
An alternative that allows you workaround the problem is to customize the Pie Radius using, perhaps, the height of the Chart rectangle to govern the dimension you need.
Eg:
procedure TForm9.Button1Click(Sender: TObject);
var cHeight : Integer;
begin
cHeight := Round((Chart1.ClientRect.Bottom - Chart1.ClientRect.Top) * 0.80); //80%
series1.CustomXRadius := cHeight div 2;
series1.CustomYRadius := series1.CustomXRadius;
end;