I can't find any service that posts the cards directly to the end recipient. Asda's website lets you add Cyrillic text to their cards but they'll post the cards to you:
http://www.asda-photo.co.uk/about/CardsHelp
However if you're going to do that anyway (and you've got a good quality printer) you might as well do what I do and design your own cards. You simply need a free DTP program, such as Serif PagePlus Starter Edition:
http://www.serif.com/desktop-publishing-software/?mc=FSSPAGEPLUS
There are some templates available but I prefer to just start with an A4 page in landscape format. Then I use the right-hand half of the page for the cover design (using artwork or photos from the web). Anything I want on the back of the card goes into the left-hand half of the page.
Once that's printed onto A4 photo paper I simply fold it into two and the outer part of the card is complete. You can't print onto the back of most photo papers, so I buy A5 adhesive stickers (with two on an A4 sheet) and print onto them, so that I can attach the stickers to the insides of the cards.
Then all that's left to do is to pop the cards into a C5 envelopes, address them and post them.