What would you do?
I'd most likely hand in my notice working in such an environment.
Is there any reason why this has to be overengineered? Not in my opinion.
I'd simply create one test case and trust my coworkers. They should be able to communicate when it's needed.
Of course getting access to all three systems is a priority (it is really necessary in fact)...
What about the three of you sit down and... work together? You can simply plug one computer in a screen and you all test it together.
(No sarcasm, seriously)
Two of the tags you posted are "HP-ALM" and "Jira".
It can be a guess, but I feel that you may be limiting you way of working because of how these tools are setup.
From the Agile ...
TL;DR: Jenkins supports pipelines from code. Make your build, test and deploy pipeline in code and you are not dependent on a specific CD tool. If you can do it from the command line and with API's you can combine and chain any tool or test-step. Use Pipelines as Code!
Is it possible to automate this complete process?
Probably, but not out of the box with ...
So I have been using a test management tool (Kualitee) that allows me to somewhat managed the same with the help of test scenarios and test cases group into one.
For example, I choose a module to test and create three different test scenarios to be executed by three different testers, I group the corresponding test cases according to scenarios and then ...