What if now, it is in our current world, and time begins to shift
slowly? When would we start to notice?
Immediately. Time shifting would mean that frequency of everything, from visible light to Internet carrier signals, would "drift". A small drift to and fro is expected due to system imperfections, but a continuous, monotonic drift would soon (a few seconds) exceed specifications and be noticed.
UPDATE: this, right here, is one of the biggest troubles for this scenario. Because the frequency of the light from the Sun would no longer match. Faster timebands would see it red-shifted, weak, unable to sustain photosynthesis and life. Slower timebands would see it shifted towards hard ultraviolet, a scorching eye in the sky. Even a small shift would change atmospheric penetration spectrum and wreak havoc with climate.
How would we palliate this and how long would it take to adapt?
It would depend on the entity of the phenomenon. Changing frequencies is doable; a continuous adjusting of said frequencies would involve redesigning long-distance communications. Still doable, though. A rapid changing would be difficult to follow, communications would be disrupted, chaos would ensue.
For significant ratios - anything more than 1.05:1, I'm afraid - you'd be looking at ecological catastrophes. The Earth is still one, it still revolves around the Sun and spins on its axis at the same speed everywhere, otherwise it would just rupture.
So, with a ratio of 1.05:1, every year the slow area would find itself 5% of a year behind. After ten years, that amounts to six whole months and the seasons would have reversed. This is likely to wreak havoc on agriculture.
More than that, 1.05:1 means that every day you lose 5% of a day, and in ten days you'll be awake in the middle of the night. People from misaligned countries will be permanently dazed and jet-lagged, or will have to live with artificial lights.
At much higher time rates, you won't get long enough nights (or days) for some biological processes. At slower time rates, you'll get very long days and nights. Both will be equally disruptive.
The stock market. I do believe that having different time response would either leave out slow country or make it something completely different altogether
Probably not so much. Different stock markets would deal with different time bands, and cross-market investing would still be possible for some financial products. You can still sell against a future date, even if it's nearer to your competitor than it is to you. That date will come for both, and both investments will be then in the past. Living faster wouldn't be an advantage in that case. I agree that fast transactions in the fast bands would probably be off-limits to the slow bands.
On the other hand, tying up a capital in the slow band could be doable while in the fast band the same period would be ruinous.
Research and high-end developpement would probably be in fastest countries. I also wonder how many people would leave the slow-scale country to move to the faster ones.
They would probably go both ways. In slow countries, you feel you live longer. If the time ratio is 2:1, you'll see a new IPhone model every six months, and your life (measured in fast years) will be far more than a century.
I have a feeling low-scale country would be very out of the political game? War would be very unbalanced for example
Not necessarily: to invade a slow-band country you'd need to go slow too. The only advantage would be in amassing forces at the time border. Consider that any campaign would last twice as much in the faster country, and be a proportionally major drain of resources.
A big trouble will be demographics. Fast countries, from the standpoint of the slow countries, will experience a demographic explosion (or implosion).
All that said...
I remember a story - I think it was in Asimov's Great Stories - about the people of the world suddenly split in different temporal lines without rhyme or reason. This is very similar to your scenario, and pretty extreme. Fast people start pulling pranks on slow people which they perceive as immobile statues; slow people feel disoriented while unseen presences rearrange their surroundings (this indicates a time displacement ratio of more than 25:1). The main character, in the end, wonders about these same questions: how is it possible that the Sun is rising in the East at the same speed for all the Lines?
(Unfortunately I read it a very long time ago in Italian, and my quick back-translation of some key terms has way too little googleness).
A viable alternative?
Since we're interested in the cultural/social aspects of time banding, we can try and come up with something that maintains most of those aspects without stretching physics too much.
Having a whole volume of space timebanded is, as we saw, not going to work too well. Perhaps different planets could work that way (and time on different planets is already running at slightly different speeds, depending on the depth of the local spacetime continuum in respect to the Sun's and the planet's gravity well. Time runs very slightly faster on the surface of the Moon when compared to Earth).
But we can try and accelerate living beings. Or just men. This has a biological impact: warm-blooded living organisms generally require thermal homeostasis, and "living faster" generates a lot of heat, while the surface to dissipate it remains the same. That's the reason why large animals tend to have slower metabolisms.
In a colder climate, a human being could live some 20% faster without overheating. Now, instead of living normally inside a faster timeband, he's living faster in a normal timeband. All problems with timeband conflict disappear; most of the "human" angles do not.
And if the acceleration only involved, say, thought processes, we could get off with even higher accelerations. Faster-reacting people would be more economical of movements, so they could also perform much faster without really moving that much faster, which would need proportionally more strength (and unlikely muscle and bone resistance). They would live, learn, research, faster. They would need to eat more and they might or might not be fertile at an earlier age, and have shorter pregnancies, giving the same demographic problems of fast timebanding.
Proportionally, they would also age faster (but again, not proportionally faster). Or they might age at a normal rate, which would give them proportionally more productivity on a per-life basis.
Depending on the causes and other side effects, all sorts of scenarios could follow. The Fasts could be the result of genetic engineering, the scions of rich people who bought better chances for their heirs (see for example Beggars in Spain by Nancy Kress). They could be the result of some (accidental) discovery about the source of life's "clock" (there's a story from the dawn years of science fiction where this was posited to be the natural background radiation, eliminating which, living being entered into stasis). It could be the result of a drug - Achilles' choice: live long, or live brightly (there's something in Ender's sequel, the Shadow series by Orson Scott Card, called "Anton's Switch").