If you buy, the LKK are sent right away into your wallet. Sometimes it can take a little bit longer because of the BTC-Blockchain: https://twitter.com/LykkeCity/status/785755823754674176

...but I believe it's faster again today. And incoming transactions are visible under "history" in the APP.

The ICO will end in about 5 hours and the market will take over right after. With other words: "Launch" will be today ;-)