The Channel Islands are part of the United Kingdom, well last time I checked they where, so BBC iPlayer and ITV Player should work in the CI, and the BBC and ITV certainly broadcast in the CI.
What you could try though, go to a website that tells you your location and country. Something like speedtest.net or mybrowserinfo.com and if it doesn't say "United Kingdom", then they probably won't work. If it does say "United Kingdom", you might want to contact the BBC and ITV and see what they have to say about it.
Maybe they got you down as been in France, because it's closer?
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