How can I use odometry + AMCL?

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The first issue I have is that I hardcode the initial odometry position at the very beginning to be at 0,0. Code is very simple and typical:

int main(int argc, char** argv){
    ros::init(argc, argv, "odometry_publisher");

    ros::NodeHandle n;
    ros::Publisher odom_pub = n.advertise<nav_msgs::Odometry>("odom", 10);
    tf2_ros::TransformBroadcaster odom_broadcaster;

    // API

    double x = 1.0;
    double y = 6.0;
    double th = 0.0;

    double vx = 0.0;
    double vy = 0.0;
    double vth = 0.0;

    ros::Time current_time, last_time;
    current_time = ros::Time::now();
    last_time = ros::Time::now();

    // Variables
    geometry_msgs::TransformStamped odom_trans;
    tf2::Quaternion q;
    nav_msgs::Odometry odom;

    ros::Rate r(50.0);

    while(n.ok()){

        ros::spinOnce();
        current_time = ros::Time::now();

        double dt = (current_time-last_time).toSec();
        double delta_x = (vx * cos(th) - vy * sin(th)) * dt;
        double delta_y = (vx * sin(th) + vy * cos(th)) * dt;
        double delta_th = vth * dt;

        x += delta_x;
        y += delta_y;
        th += delta_th;

        // Setting the transform stamped messages:
        q.setRPY(0,0,th);
        odom_trans.header.stamp = current_time;
        odom_trans.header.frame_id = "odom";
        odom_trans.child_frame_id = "base_footprint";

        odom_trans.transform.translation.x = x;
        odom_trans.transform.translation.y = y;
        odom_trans.transform.translation.z = 0.0;
        odom_trans.transform.rotation.x = q.x();
        odom_trans.transform.rotation.y = q.y();
        odom_trans.transform.rotation.z = q.z();
        odom_trans.transform.rotation.w = q.w();

        // Send the transform:
        odom_broadcaster.sendTransform(odom_trans);

        // Send the odometry messages to ROS:
        odom.header.stamp = current_time;
        odom.header.frame_id = "odom";
        // Here, take the positions from the wheelme backend whenever it reaches and send it to ros:
        // Replace the xyz values by path values in for loop
        odom.pose.pose.position.x = x;
        odom.pose.pose.position.y = y;
        odom.pose.pose.position.z = 0.0;
        odom.pose.pose.orientation.x = q.x();
        odom.pose.pose.orientation.x = q.y();
        odom.pose.pose.orientation.x = q.z();
        odom.pose.pose.orientation.x = q.w();


        // This values will be always zero until wheelme will start publishing its velocities:
        odom.child_frame_id = "base_link";
        odom.twist.twist.linear.x = vx;
        odom.twist.twist.linear.y = vy;
        odom.twist.twist.angular.z  = vth;

        // Now publish all that to odom topic:
        odom_pub.publish(odom);
        last_time = current_time;
        r.sleep();

    }

}

After that, I try to initialize the AMCL using my Lidar and I can see that the position of the robot jumps between the harcoded inital odom values and predicted AMCL values.

How can I combine them both? So that the AMCL corrects the position of the robot and odometry frame accepts that change?

EDIT: adding a tf tree