So is the goal that the very first thread to complete will change the URL for all subsequently created threads?
My understanding of the documentation is that you can't change the value of a property inside the thread-group:
Properties can be referenced in test plans - see Functions - read a property - but cannot be used for thread-specific values.
(see http://jmeter.apache.org/usermanual/test_plan.html#properties)
My assumption is that each thread in a thread-group gets a copy of the properties. If you change the value of the property inside the thread group, then you are actually changing the copy for that particular thread. Since you are changing it in the post-processor, the thread is very likely just about to be disposed, resulting in your change being lost. After disposal a new thread is created but with the original value of the property.
So what you need to do is figure out how to change the value outside of the thread-group.
I have done something similar in my own tests whereby I am changing the value of a property in the middle of the test, and the value is picked up immediately by all of the active thread-groups, resulting in each new thread created from that point forward getting the new value. I am doing this by using the Beanshell Server: https://jmeter.apache.org/usermanual/best-practices.html#beanshell_server
In my specific case I use jenkins job that calls a shell-script which connects to the beanshell-service running on the local-host:
java -jar ${jmeter_home}/apache-jmeter-5.0/lib/bshclient.jar localhost 9000 ${test_plan_home}/update_Prop.bsh "${property}" "${value}"
where my update_prop.bash file is simply:
import org.apache.jmeter.util.JMeterUtils;
JMeterUtils.getJMeterProperties().setProperty(args[0],args[1]);
You would not need to use Jenkins or anything like that, though - if you setup your JMeter Process to include the Beanshell-server (see the link above) then you can simply replace the code in your post-processor:
${__setProperty("url", "google")};
with the code to connect to the beanshell server and execute that command there instead:
exec("./updateprop.bash url google");