SQL Server Performance hit after cache rebuild
So, will the SQL server do this immediately after my stats are updated or will it wait until a corresponding query is executed to rebuild the cache ? 


NO, SQL Server would not do it immediately. Rebuilding indexes does lot of page movements and so does stats update. There is quite a chance that pages already present in memory, before index operation, would have been thrown back to disk. Now when you run the query again SQL Server optimizer will prepare new plan looking at new statistics and would request I/O to bring pages into memory which were thrown back to disk. 


  Will there be any slight performance degradation since it has to build the cache from start again before executing the query ?


This can cause a bit of more time being spent by optimizer in preparing new better plan and I/O bringing pages into memory but this would just be the first time. But there would not be much performance degradation. In many cases the new plan would execute query much faster and this would outweigh the time taken by optimizer to build new plan and I/O to get pages into memory. 

Since you are interested about plan cache I suggest you read Plan Cache and Reuse. As per the document among other following two can also cause plan to recompile


Updates on statistics used by the execution plan, generated either
explicitly from a statement, such as UPDATE STATISTICS, or generated
automatically.
Dropping an index used by the execution plan.


BUT note that plans do not always get invalidated when you update stats. There are quite few articles describing this


What caused plan to go horribly wrong
Can update stats cause recompile when no data has changed