It is unlikely that there are benefits for healthy individuals. On the one hand, the amount of water one should supposedly drink each morning in this therapy, would lead a person to nearly exceed their recommended daily intake, all in one go. On the other hand, the magical claims posited are already mostly shown to be myths.
This article from The Guardian explored popular claims about how much water people should drink. The article contains a neat summary of references from a review article in the British Medical Journal as well as interviews with kidney specialists and general practitioners.
About the amount of water needed per day:
According to the European Food Safety Authority, healthy adult women need two litres a day, and men around 2.5litres. We get around 20% of our water from food.
The above quote implies that as a woman, I get about 400 ml from food (men get about 500 ml), and drinking 1.5 litres would mean that I should take only 100 ml of water or other liquids apart from this morning dose. As kidneys work around the clock, it is likely that a woman would either have to be thirsty most of the day, or to exceed the recommended daily dose of water, if she were to drink 1.5l in the morning. Neither of these options are likely to be healthy. (Men would need an extra 0.5l during the day - so perhaps the 1.5l in the morning is less of an issue for them.)
About the magical outcomes of drinking more water than your body tells you to, some of which are present in your first link:
(...) Stanley Goldfarb, professor of medicine and a kidney specialist at the University of Pennsylvania (...) reviewed the scientific literature on the health benefits of drinking a lot of water, identifying the four recurrent themes that were put about by those who advocated it.
"One was that water improves your skin," he says. "We showed there was no scientific basis for that.
The second myth is that drinking water is an aid to diets and would reduce your appetite. That has been carefully studied and it doesn't. If you flavour the water, that will suppress your calorific intake during the subsequent meal, but nobody has shown that it suppresses it over 24 hours.
The third myth he looked at is that drinking water flushes more toxins out of your body. "All it does is increase the volume of your urine, but it doesn't change the material in the urine.
The last issue that people have advocated is that water can control headaches. It was not substantiated."
In sum, drinking lots of water has not been shown to lead to the health benefits that water therapy claims it should have. This research does not cover specifically drinking water in the morning, but since water therapy relies on ideas that are wrong to begin with, it is most likely not beneficial to people's health.
