I'd love a 5-axis machine for £15k!

It is one of those parts that although you think would be simple to make, is actually quite difficult.
I'd go for Jazzs suggestion of get them done on a 3axis, but get the tapers stepped/waterlined, then finish on a sander. A jig on a decent (i.e. a floor standing one, not a stupid handheld thing!) belt or disc sander will remove the remaining material pretty quickly. Might not be the quickest or cleanest method, but it'll do the job.