Update statements with set transaction isolation level read uncommitted
If I have a stored procedure with set transaction isolation level read
  uncommitted, will it affect update statements?


Read uncommitted allows dirty reads. An X lock will be taken on the row or higher level (in the data page or index) before it is made dirty. Rows accessed directly by the query when locating a row to update will take a U lock and be blocked.

However it is still possible for an Update to be affected by the isolation level.

Connection 1

CREATE TABLE T1
(
X INT NULL,
Y INT NULL
);

INSERT INTO T1 DEFAULT VALUES;

BEGIN TRAN
UPDATE T1 SET X = 100;    

WAITFOR DELAY '00:00:10'

ROLLBACK;


Connection 2 (run this within 10 seconds of firing off connection 1)

SET TRANSACTION ISOLATION LEVEL READ UNCOMMITTED;

UPDATE T1
SET    Y = (SELECT SUM(X)
            FROM   T1);

SELECT *
FROM   T1; 


Result



The read operation read the uncommitted value and the final result was used in the UPDATE even though the 100 that was read was eventually rolled back.