"Start" vs. "begin", "finish" vs. "end"

Solution 1:

I would go with simple past on all of those. It doesn't matter whether you use began/started or finished/ended, since they are interchangeable.

The year in which physiotherapy treatment began.
The year in which physiotherapy treatment ended.
The year in which physiotherapy treatment started.
The year in which physiotherapy treatment finished.

Note that you could simplify the sentence still further by removing unnecessary words:

The year the physiotherapy began.

Or even:

The year I [she, he, they] began physiotherapy.

Solution 2:

I think begin sounds natural here. I'd prefer using start when I do something on my own, for example, "I started my scooter."