As an update, I made the vases using an augur bit in the end, but I had to finish the taller one with a normal bit, to get the full depth for the test tube that goes down the center of the vase.
Took a few minutes to tune the auger bit in, bit once that was done, and with judicious use of the oil in a can, I was able to bore the holes with no real bother.
I had thought I would have to use a power drill for this (the auger bit is a hex shank), but after grinding down the shaft of a 3 jaw chuck spindle, I was able to do the boring by hand.
Apologies if the image looks a bit strange, I am not sure how well the whatsapp image will upload.

Next challenge 🙂 I have been asked to make a larger version of the vase, freestanding on the floor, roughly half to three quarters a meter or so tall, and would be made from a section of branch, and I’m thinking that for stability it would have to be fairly wide diameter. Now, that would mean that for the tube to hold the flower not to look way out of proportion to the body of the vase, that I would need a 40 – 60mm diameter hole.
This is the challenge, because as far as I can see, most long spiral bits are not available in such a large size. Forstner bits do, but they are fairly short shanked. I can probably cobble together an extension rod for the forstner if needed.
Or I can use a hole saw, drill a bit, clear the plug, drill some more, clear some more.
Any other ideas on how to do this?
(Splitting, gouging, and then reassembling is also an option, but I am not sure how that would look in the end though.)
Colin.