The German "Lasst es Euch schmecken" or "Lassen Sie's sich schmecken" (familiar/polite) is used in a casual way instead of "Guten Appetit" or "Mahlzeit (casual, regional)" at the beginning of a meal.
To "tuck in", in American English, is commonly used to tuck somebody's sheets in at bed time. In British English, it is indeed also a colloquial expression to encourage somebody to start eating. A similar sounding equivalent to "Lasst es Euch schmecken" could be "dig in" instead:
[ in imperative ] (dig in) informal used to encourage someone to start eating with gusto and have as much as they want: put the sausage on top of the polenta; then dig in. (From the New Oxford American Dictionary)
On the other hand, the American English "dig in" is more similar to the German "greif zu" than to "Lasst es Euch schmecken".