I can’t see your dimensions, but there are standard
industrial/medical products that are already halfway along to what
you need, making re-fabrication to perfect size an easier task for
Filter funnels and powder funnels from laboratory suppliers (merely
an example http://www.livingstone.com.au ) are starting points in
glass, which have the conical shape, are open at the bottom but
easily pinched off. Large glass test tubes are already pinched off,
but not conical.
If you want to go the plastic route, you could start with clear
acrylic tube, melt and pinch one end, use low pressure compressed air
on the far end with a hot air gun to soften the necessary part of the
length, using your existing metalwork as the form. You can
lathe-finish the drinking lip.
For the full professional thing, a chemical glassblower would be
your best contact - but I suspect you can have fun with a funnel or
two, a couple of test tubes, a length of acrylic tube, first. What’s
a couple of burned fingers if you gain the satisfaction of personally
crafting the flute?