I have found that when a worker moves into a hex and starts work having used a part of their movement points, they will start work on the end of that turn, not the next turn.
This is similar to when a military unit heals on a hex and that they do not start healing unless they end the turn with all their movement points.
If a worker starts work when they have used a movement point (or percentage of) do they only put that percentage of work into their turns remaining?
For example, if a worker starts work when they have 1 out of 2 movement points remaining, do they only put in 50% which is half a turn of work?