There used not to be any such thing as cryptic crosswords in AE either, it was a BE thing. But they're so good that their popularity has spread, and now there are solving hints and even paperback books explaining the different forms a clue can take (two meanings of the word, two parts of the word, anagrams, word hidden in the clue, odd/even letters, first/last letters, etc.). If you otherwise like crosswords and wordplay, it's absolutely worth it to give yourself a little boost just in learning the system. In the US, the novelty puzzle in the Wall Street Journal weekend edition is a cryptic one about once a month.
The trick in solving authentic British ones (which I also find much harder than the Brits do) is also to know a lot of cultural stuff like abbreviations and slang. There, too, you just partly pick it up by doing it, even if when you start out you have to look up a lot of things. Look for ones where in the solution, they hint at how they arrived at the answers by breaking them into pieces.
German does sometimes have puzzles similar to cryptic ones, even if they may not stick to the 'rules' of forming a clue quite as strictly. I've probably told this before, but I remember a trip over a decade ago where our little train was delayed briefly due to a report of a thief (!), and a very nice Swiss-Italian couple helped me, first with understanding the announcement (see #6), and then with a cryptic-like puzzle from, IIRC, Der Standard.