@SakshiBagal 

ACL won't come into picture as you are using GlideRecord and not GlideRecordSecure.

Your BR condition should be this

Assigned To [Changes] AND Assigned to [IS NOT EMPTY]

Things to check

1) did you add log and see if query ran?

2) are you using correct source field to query and correct acknowledge field?

var gr = new GlideRecord('on_call_escalation');
gr.addQuery('source', current.sys_id);
gr.query();
while (gr.next()) {
    gs.info('Record found');
    gr.acknowledged = true;
    gr.update();
}

If my response helped please mark it correct and close the thread so that it benefits future readers.