There is no easy answer to this. It depends on what you are going to do with the company and - if applicable - if there are any DTAs you can utilize to minimize your own tax-burden. For example, while you may be paying 0% tax in Cyprus, you may still be liable for taxation in the country where you are domiciled, whereas paying UK corporate tax may be the only tax you have to pay if there is an adequate DTA.
In terms of anonymity, neither jurisdiction offers high privacy as company details are on public records. You would have to use for example nominees to keep your name out of public records.