Database synchronization time in cassandra
Cassandra provides tools and insights of all this internal information with the nodetool gossipinfo command and cqlsh tracing.
In the scenario that you are proposing, I'm inferring that you are using a Replication Factor of 2, and that you are interested in the exact time that is taking to have the information written in all the nodes, you can measure the time required to do a write with the consistency level set to ALL, and compare it with similar writes using the consistency level of ONE. The difference of the times will be the propagation from one node to the other.
Finally, if you are interested in measuring the performance of the queries in Cassandra, there are several tools that enhance the tracing functionality, in our team we have been using zipkin with good results.