My wild ass guess is one month.

I base this number on marijuana testing supposedly useless after a month. It does work very well for two weeks though because some byproduct is fat soluble and is absorbed by your fat cells where it takes them up to a month to completely rinse out. So, for example if you were eating a lot of high hormone ranch animal meat there may well be junk in there that takes your system a month to completely purge it out after you stop eating the high hormone ranch animal meat and you would not be able to see the effects undiluted for a month. Also this is the time granule I use for my own diet self-experiments which I seem to never get to the end of. My diets are for fitness and health purposes only as I have never been much over- or underweight.

What is the minimum effective period over which one should try a new dietary plan, before reaching conclusions on its effectiveness?

Two weeks. But don't draw conclusions based on, for example, weight even if 'looking like you have lost weight' is actually your goal.

This question could be generalized to other health issues where self-experimentation is appropriate.

It could... but the more generalized the answer the less useful it is going to be. Specifics are kind of the important thing here!