I had to pass ownership of testing to Tester B for Fix 1. Once Fix 1 is released to production, you realise Fix 1 is not being tested properly and it affects the productivity on the consumer end. What’s the best approach as Software QA Tester to this situation?
For me, it wouldn't make any difference if you did all the testing, B did all the testing or if you shared the effort. Nor does it make a (big) difference if what you notice in production is a new problem, an incomplete fix or a regression.
In all cases, you should follow the procedure for reporting and assessing a problem found in the production environment.
The only way in which the type of bug makes a difference is if you do a postmortem analysis on why the problem got released into production and how you can improve your processes (as a team/company) to prevent that from happening again.
You need to define further what you mean by, "...is not being tested properly". Was the testing not proper because the tester improperly performed the test or was it improper because the test was inadequate? Without an answer to the root cause, you have no possible action. So to answer your question, your best approach is to uncover the root cause. Then that will inform your next steps.