Expression for "The time period in which I started working in my current workplace"
“When I first worked at X” seems to fit your need in your second example.
The first example has not quite the same structure. It might be expressed as “James first worked at X from 20th ...” or “... first worked at X after 20th ...”
I prefer after to since because since is often used to imply that whatever started at some time is still happening. And that is not necessarily the case here: James or I may have left the job.