Read Committed Snapshop Isolation Level - Effect of NOLOCK
If I understand correctly, a query that uses WITH (NOLOCK) changes the
  isolation level. When this is used in a query, is it now possible for
  a reader to block a process attempting to update the data being read?


No, you think wrong.
Both in RCSI and Read Uncommitted readers don't block writers.

The mechanism is different: RCSI uses last committed copy of modifying data for the reader, and a SELECT using with (nolock) will access "dirty" data, but there is no blocking in both the cases.

Here is an example. You make an update like this:

begin tran
   update dbo.t
   set col1 = 20 where col1 = 10;
-- transaction is still open, X lock is held on a row


In the second session you make SELECT * from dbo.t;

In the case of RCSI you'll get back col1 = 10, in case of Read Uncommitted you'll get col1 = 20, but session 2 will return data in both cases immediately, without waiting when transaction from 1st session commits/rolls back