OK, guys, it took about 40 minutes in total, as I had the materials to hand.
Yellow felt (though as I only had the cheapy rainbow pack of felt from Tesco, I only used 1 shade of yellow instead of 2)
Green felt
Green pipecleaner
Yellow and green thread
Dab of superglue or PVA

If I may apologise at this point for the quality of the photos - took them on my phone and it's really rubbish.

Stage 1:

Cut your strips of felt to size, and snip a deep fringe in one side. DO NOT cut too far down, as this interferes with the structural integrity of the strip when you're wrapping it round.
If you're using 2 shades of yellow felt, the darker shade goes in the centre of the flower, lighter on the outside.

Here's my first yellow strip, out of focus but suitably snipped:


Stage 2:

Now get your pipecleaner - the original maker wrapped the pipecleaner with wool... I happened to have a bright green one and really couldn't be arsed with all the wrapping. Do as you wish with your pipecleaner. Then get the first yellow strip, snipped into fringey dandelion petals, and glue it to the last half centimetre of the pipecleaner. Really do glue it, I didn't and spent the entire build having the stalk depart at unexpected moments...



...like this...



Stage 3:

Next up, wrap the first yellow strip tightly round the pipecleaner, and stitch in place using yellow thread. That is, stitch the felt to itself, not to the pipecleaner. Now do the same with the second yellow strip. One that was done, I stitched quickly round the bottom edge of the whole flower, securing several layers together with each stitch, as I could foresee unravelling-petal problems in the future if the glue happened to give way (say, if it rained - I know it's a remote possibility in Scotland, but still!). The photo is a bit fuzzy but you can see the dimpling from the stitches.



OK, so your petals are done. Congrats, you're half way there! If you 'fluff' the petals out by flicking them with with thumb and forefinger while holding the stalk, they should look like this:



Stage 4:

Now it's time for the underneath of the flower. This is apparently called the calyx, and in our case at least, is composed of two small strips of green felt. The first, which makes the downward-curving part, looks like this:



Just cut a narrowish strip of green felt, and wrap it round the base of your petals; this will give you the length you need. Then just snip the little curves in as if you were making a collar for Kermit the Frog.

When the bottom strip is cut, sew it round the bottom of the petals with the points pointing downwards, like this:



Stage 5:

The original maker of these felt dandelions was a bit vexed with her design for the upward-pointing bit of the calyx. I took a look at the real dandelion next to her felt one, and came up with this:



Measure the green felt strip as for the previous one, and snip irregular pointy blades into it, as if making a fuzzy felt lawn. I found this (as per the pic) worked OK, but it was a bit too regular and pushed the petals upwards too firmly, so (unpictured, I do apologise) I simply went along the strip and snipped out a blade every now and then, making a much more irregular array. I think this looks more natural.

Once the upper calyx has been wrapped around, stitch into place. It should look more or less like this:



You're finished, or almost - I clean forgot about cutting out and attaching the dandelion leaf, but that's easy peasy.
The end result of all this hard work:



Ta-dahh! Though why Photobucket and the forum software refuse to put it the right way up, I'm not sure.

I'm pleased with it, though I do say so myself. And it's super-easy as well, which just adds to the pleasure.

Have fun!