In short, the best way by far to heal your settlers in Going medieval is having them sleep whatever ills them off.
However, just sleeping on its own is not enough to completely rid them of an ouchie boo boo or wound. This is because you will also need someone assigned to tending wounds for them to be healed by this process.
That being said, you can assign one effectively to tending wounds by visiting the job tab at the top left of your screen. Once this tab is open, find a settler with a start or a high level in the tend skill, and then click the number until it is one.
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This will then set tending wounds the highest priority for that settler, but this will only work if that settler is in bed. So to force a settler to go to bed so they can be healed, you will also need to set their convalesce job to priority one. Doing so will effectively force them to go to bed to be healed whenever they finish their current task.
Also, if your settler is in a critical state, you can prioritize healing them by selecting another settler and then hitting prioritize healing wounds over the wounded settler.