At last, I think I have cracked making reasonably convincing eyes.
This is how...
Download a load of photos of human irises of various colours. Resize them to the desired dimensions and print them off a decent colour printer (ta muchly to my old mucker Joe Scaramanga). Fortunately, the size iris I needed coincided with a single-hole punch I have, thus ensuring perfect circularity - trying to cut them out using scissors proved a dead loss...
Take a 16mm spherical white plastic bead and flatten one face such that the diameter of the flat matches the diameter of the paper iris. Stick the iris to the flat face (I used doublesided tape, which I stuck to the back of the iris before punching it out).
IMPORTANT NOTE: Since originally writing this post, I have found that it is not necessary to flatten any part of the spherical bead - the paper iris can just be stuck to the curved surface, it is small and thin enough to conform to the compound curvature without distortion or wrinkling.
Now, take another bead of the same size and use it to vacuum form a clear dome using thin transparent plastic sheet. If you don't have a vacuum former (I happen to have made one twenty-odd years ago for some other projects) you could press-form the domes: take a piece of scrap wood, drill a hole a few millimetres larger than the bead, tape/pin your clear plastic over the hole, stick it under the grill until it goes soft and then press the bead into it.
Glue the dome over the incomplete eye to finish it off.