Once again, it is not all about the grammar.
in the next three weeks, in the coming three weeks, or in the following three weeks confusing.
Yes it is confusing because we have isolated these three phrases and we therefore have no information to support us as we try to understand the meaning of each. What is the context? what is the location? in your examples we have no indication therefore to give a definitive answer is not possible.
in the next three weeks
The first example is possibly the easiest to explain, which is fortunate, as it is probably the most commonly used of the three phrases.
If this is used in a general context it vaguely starts from some time now. If it is in work environment where the company follows a strict working week schedule it starts from the first day of the next working week.
For example a logistics company would need to be very precise when relating to time schedules where as a pet shop possibly does not need such a precise schedule.
next; adjective, pronoun; being the first one after the present one or after the one just mentioned: RefC.E.D.
in the coming three weeks,
The second example
This is a vague context and means something is happening soon and of course, soon is a relative word.
coming; adjective [ before noun ]; happening soon: Ref C.E.D.
Having said that, with all your examples, it also depends on the topic of the conversation and therefore the context of said conversation. This is particularly true in your third example as we have been left dangling off a cliff!
in the following three weeks
Following what? Do we make an assumption following has been used instead of "the next"? or has there been something previously mentioned that we can relate the start to.
following; adjective; [not gradable] (NEXT) Ref C.E.D.
coming or happening after something else in order or time:
Here is an example of how following could be used
I’m busy on Thursday, but I’m free the following day.
So for your example I would expect something like this
During the first two weeks of July the factory will be closed and in the following three weeks we will be really busy.