A passport is not a National Identity Card. This is an entirely different document that citizens in many EC countries have to carry at all times. What that site is saying is that if your passport has less than three months to run, but you have a separate id card you'll be OK. We don't have National Identity Cards in the UK so I reckon you'd better get them renewed quick.
It will be the check-in at your departure airport that decides whether you go, not Immigration in Cyprus. The airlines get fined if anybody gets refused entry, so they are very careful who they let travel.
I renewed my passport earlier this year and it took just 4 days.
You can do it online here:
http://www.passport.gov.uk/passport_index.asp
Have a good holiday, cheers