There are no rules for estimating time it takes to write tests for a use case. But it improves as you progress. Some code may be hard to test. That's a sign that the code is not modular enough. If you refactor the code to fix its coupling problems then the next similar code will take much less to test. Needless to say you can write reusable test  base classes which you can derive from in many cases. So over time it takes less and less if you refactor mercilessly your code and tests.

And I haven't seen any reliable case studies showing how much you save time later on.