Cosmote certainly do online billing. I simply get an email with the bill attached. You can then go to their payment page
enter the payment code on the bill and it should take you to a page showing the amount owed. Tick the box accepting terms and conditions and click on "Submit". That takes you to a typical card payment page where you fill in your card details.
I have no idea if it will accept a UK card. I only use Greek accounts and cards for transaction in Greece and UK accounts and cards for transactions in the UK. I suspect that IF there is a problem it will be that one of the items they use to authenticate your card is your address. The bill will be registered to a Greek address and they will probably expect the card to be registered to the same address. If the card is registered to your UK address then the two don't match and the payment might be rejected by the credit card provider. I am not saying that this will be a problem but it might be. If paying with a UK card previously worked for Vodafone I don't see why it won't work with Cosmote.