Currently, autopropagation in Trados Studio with Language Cloud enabled takes a long time - up to several minutes to propagate 20 segments on a 7000 segment file with a 10 mbps connection (somewhat faster on a faster connection) but still too long considering that with Language Cloud disabled autopropagation takes under 1 second on the same file.
Testing was done today with Alex Jude (RWS support expert) under Case No. 00596129. 

This means that autopragation works quickly when done locally in Trados Studio but it is the exchange with the Live Team server that slows it down dramatically.

This seems to give the clue that autopropagation algorithm is set up so that it exchanges some kind of messages with the Live Team server for EVERY segment to be propagated. 

The solution would be to treat this as a batch task performed locally in Trados Studio on the entire group of matching segments followed by a single message sent to the Live Team server (the list of segments to be propagated) to update the server. 

The situation is made much worse by the fact that Trados Studio is completely unresponsive during autopropagation, precluding any further tasks being performed.
So the second suggestion would be to make autopropagation to the server a background parallel process, so as to fully utilize the power of Trados Studio being local software and enable the translator to continue their work while the data is being transferred to the server.
An indicator could be added somewhere in the corner to show how many segments are in queue to be updated on the server to provide clear feedback.