Are professors allowed to accuse someone of cheating based on a general increase in test scores?
First, as you already noted, the better scores and the bimodal distribution may have other reasons. For example:


One of three TAs for the course did an outstanding job.
The professor reused questions from previous exams (perhaps without noticing) and one third of the students informed themselves about previous tests.


Most importantly, the following thing is odd: It’s highly implausible that the rate of students who decide to cheat on their own increases by that amount from one year to the next. Thus, they would have to decide for cheating together or due to some common motivation. This in turn only makes sense if the cheating is a collaborative endeavour or if somebody is helping the cheaters.

Now, while such scenarios are not unthinkable, they are usually as likely as possible explanations that do not involve cheating (your mileage may vary depending on the circumstances). Moreover, even such a scenario happened, just repeating the exam is unlikely to solve it. Therefore the professor should first find out what actually happened.

So, the statistics may point at something being unusual, but without further investigations you cannot say that it’s cheating. Moreover, even if it is cheating, statistics can only tell you that it happened, put cannot point to any individual involved in it.

To address your individual questions:



  Are professors allowed to accuse someone of cheating based on a general increase in test scores?
  
  
  
  The professor goes on to claim that those students who had significantly higher scores than in their previous test must have cheated! Are professors allowed to that?


Technically, yes. Almost everybody can accuse everybody else of everything as long as we are not entering the domain of libel laws. Will it be successful or a good idea? Probably not, at least in any reasonable university or jurisdiction. And that’s not even considering that he has not one but many students against him.

  is it ethical and reasonable to assume that one third of the people in his class were cheating?


Assuming something cannot be ethical or unethical; acting upon an assumption on the other hand may be, but that depends on the action.

  Is it normal practice and ethical for professors to accuse someone of cheating just by looking at the student’s test scores?


Normal practice – I never heard of it. Ethical – certainly not. There are several possible reasons for a good individual test result, e.g., hard work and plain luck. The professor should be aware of this and thus the accusation is unfounded and hence unethical.