Well, I've just wasted about 20 minutes looking through all of the FAQs, and other potentially relevant pages, on the Celebrity Cruises website but I'm still none the wiser really!
All that's there is this:
"[Q] How do I check in for my cruise?
[A] The quickest way is to use the Celebrity Cruises® app, available between 90 days and three days before your sail day. Just sign in, select your cruise, and tap on the check-in button. Make sure you take your security photo and add your passport or travel documents for the fastest boarding experience on your sail day."
The reference there to 'the quickest way' suggests that there is actually an alternative method but nowhere on their website do they actually tell you what it is!
However I have, at least, found a UK phone number for you to call: 0344 493 2043. Lines are open Monday - Friday: 09.00 - 20:00, Saturday: 09:00 - 18:00 and Sunday: 10:00 - 17:00. Calls are charged at the same rate as calls to 01 and 02 numbers are, from both landlines and mobiles. So, if you've got spare 'included minutes' on your phone package, the call will actually be free.