It sounds as if the profile may be the problem. Presumably you checked to make sure that the version of the profile you downloaded corresponds with the identical printer model you are using.
Ilford's site makes it fairly easy, through a series of drop-downs. It is possible the site is serving up the wrong profile after I make the choices, but the profile name makes sense for my printer.
One way of checking for profile behaviour would be to open the image having the cyan cast when printed, and soft-proof it in Photoshop. If you find when you activate the soft-proof that a cyan cast appears, you will have a good indication that it could well be a profile issue, because the softproof is supposed to read back to your display the effect of using the profile being soft-proofed.
If the soft-proof shows no cyan cast in Photoshop, this would indicate that the profile is probably OK and perhaps there is a setting in LR which needs to be tweaked.
Good tip. I just tried that, I cannot detect a cyan-cast when switching between the Epson PremLuster and Ilford GFS profiles. The image definitely takes on a slight warm-tone when I switch the proof setup to the GFS setting, which I would expect since this paper does have a pronounced warm tone. Hmmm. The image I get from the printer is almost a perfect match to the soft-proof when using the Epson profile and printing on the Ilford paper.