Since I moved.....

Yes, you can use two present perfect verbs in one sentence. However, in your example the "since" means from a point in the past until now, so making the first part present perfect is really overkill. It would sound better to say, "Since I moved to London, we haven't called each other."

Have ed here doesn't make sense, strictly speaking. If the case is "we haven't called", ie, some fact lasts till right now, then this fact gotta start from a time in the past. Which's to say, usually it's always the past tense that works with since. Pretending the "since have ed" one was right, the only way to comprehend it would be: Now that I have moved to London, we can't call each other. Y'know, another meaning of "since".