I found out the reason. The '_check' function in 'Check Action Rules' scheduler checking a condition before executing the process. That condition checks whether the action date greater than last automated action run datetime and less than of current datetime. Whenever the '_check' function executes it saves that time as last run date. So that's why when we run the scheduler manually by putting back date, it is not executing process.

/addons/base_action_rule/base_action_rule.py  line: 334

##For testing you can modify the code as 

if action_dt <= now: 

def _check(self, cr, uid, automatic=False, use_new_cursor=False, context=None):
        """ This Function is called by scheduler. """
        context = context or {}
        # retrieve all the action rules to run based on a timed condition
        action_dom = [('kind', '=', 'on_time')]
        action_ids = self.search(cr, uid, action_dom, context=context)
        for action in self.browse(cr, uid, action_ids, context=context):
            now = datetime.now()
            if action.last_run:
                last_run = get_datetime(action.last_run)
            else:
                last_run = datetime.utcfromtimestamp(0)

            # retrieve all the records that satisfy the action's condition
            model = self.pool[action.model_id.model]
            domain = []
            ctx = dict(context)
            if action.filter_id:
                domain = eval(action.filter_id.domain)
                ctx.update(eval(action.filter_id.context))
                if 'lang' not in ctx:
                    # Filters might be language-sensitive, attempt to reuse creator lang
                    # as we are usually running this as super-user in background
                    [filter_meta] = action.filter_id.get_metadata()
                    user_id = filter_meta['write_uid'] and filter_meta['write_uid'][0] or \
                                    filter_meta['create_uid'][0]
                    ctx['lang'] = self.pool['res.users'].browse(cr, uid, user_id).lang
            record_ids = model.search(cr, uid, domain, context=ctx)

            # determine when action should occur for the records
            date_field = action.trg_date_id.name
            if date_field == 'date_action_last' and 'create_date' in model._fields:
                get_record_dt = lambda record: record[date_field] or record.create_date
            else:
                get_record_dt = lambda record: record[date_field]

            # process action on the records that should be executed
            for record in model.browse(cr, uid, record_ids, context=context):
                record_dt = get_record_dt(record)
                if not record_dt:
                    continue
                action_dt = self._check_delay(cr, uid, action, record, record_dt, context=context)
                if last_run <= action_dt < now:
                    try:
                        context = dict(context or {}, action=True)
                        self._process(cr, uid, action, [record.id], context=context)
                    except Exception:
                        import traceback
                        _logger.error(traceback.format_exc())

            action.write({'last_run': now.strftime(DEFAULT_SERVER_DATETIME_FORMAT)})

            if automatic:
                # auto-commit for batch processing
                cr.commit()