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You always get self.connections == 0 because you don't update the value within your while loop. But this is not the main issue with your code.

How I understand you question is that you want to have a node that subscribes to a topic and stores the current heading if a service is triggered and afterwards provides this selected heading to all nodes that subscribe to a second topic. In this case, your code could look like this:

A latched publisher automatically sends its current message to any new subscriber.

class save_hdg(object):

    def __init__(self):
        self.latch_publisher = rospy.Publisher('/autonomous/saved_heading',
                                               Float64, queue_size=10, latch=True) # TURNING ON LATCHING

        self.serv = rospy.Service('/autonomous/save_heading', Empty, self.publish_once_only)

        self.hdg_subscriber = rospy.Subscriber('/mavros/global_position/compass_hdg', Float64,
                                               self.subscriber_callback, queue_size=1)

        # initialize all member variables in init
        self.heading_received = False
        self.current_heading = self.hdg = None
        rospy.loginfo("The save heading service is ready")

    def publish_once_only(self, req):
        rospy.loginfo("save heading service has been called")

        if self.heading_received:
            self.hdg = self.current_heading
            self.latch_publisher.publish(self.hdg)
        else:
            rospy.logwarn("Did not receive heading yet, can't republish")

        return EmptyResponse()

    def subscriber_callback(self, msg):
        self.heading_received = True
        self.current_heading = msg.data

if __name__ == "__main__":
    rospy.init_node("save_heading_ServiceServer")
    save_heading_object = save_hdg()
    rospy.spin()