One other thing, the speedbrakes will raise when the reverser levers are positioned for idle reverse , which will happen not earlier than engines go down to idle thrust. This takes about 2 seconds , so you won't have automatic raise of the speedbrakes untill about 2-3 sec from deciding to RTO.
By Boeing procedure (for the 737 ) , the Cpt will retard the thrust then raise manually the speedbrakes. The engines will be at idle thrust at which time the lock will open and allow reverse thrust application.