EDIT #2: In your edited question, you have (rightfully so) shown your continued confusion, as your question wasn't really answered yet (I have added emphasis on the word "working"):
What I don't understand is since Vb>Vc now, why doesn't current flow from B to C now? If this happens how is there even something called saturation region. Why
doesn't the whole processes that happened while in the working region
stop?
FIRST, let me clear it up for you: BJT transistor NEVER STOPS WORKING as long as its base-emitter junction "diode" is forward-biased.
Your confusion stems partially from your incorrect view and from the slightly misleading naming of the transistor operating modes as active and saturation - a transistor is ACTIVE/WORKING (amplifying its base current) throughout both its LINEAR and SATURATION region!
In your mind, you may be thinking active and non-active. Our subconscious minds play such simple tricks on us sometimes, especially when our picture, knowledge or understanding of something is incomplete.
To help demonstrate my point, I have taken one of my examples from below to the most extreme case which never happens in reality but it helps you understand.
Let's assume the collector-emitter channel is so deep in saturation that it has ZERO resistance and the collector and emitter are practically shorted and at exactly the same, GROUND potential.
This would make the base-emitter junction diode and base-collector junction diode have EXACTLY the same voltages since both PN junctions will be connected in parallel.
simulate this circuit – Schematic created using CircuitLab
Even in this unrealistic worst case scenario, the base-emitter diode conducts as much current as does the base-collector diode, in other words their currents are equal and the "amplification" of the transistor becomes 1, meaning no current amplification.
But this is practically impossible because the basic theory of "transistor-action" means that there has to be base-emitter current for the collector-emitter current to flow, this base-emitter current can't be greater than the collector-emitter current, AND because the collector-emitter channel ALWAYS has some resistance.
Finally, a BJT is not just two diodes sandwiched back-to-back, but it has an additional collector-emitter channel represented as a resistor connected between collector and emitter, and its resistance depends upon the base current.
It is almost like the base-emitter is a diode whose current controls the collector-emitter channel resistance, as below.
simulate this circuit
The above schematic would be a more complete representation of a BJT.
For all of the above reasons, you will always have at least some voltage drop (at least a few mV) across the collector-emitter channel and the base-emitter voltage will never be less than the sum of the base-collector and collector-emitter voltages, even when the base-collector voltage is lower than the base-emitter voltage or even when the base-collector voltage is zero; its voltage relative to ground is still no less than the base-emitter voltage.
If the base-emitter voltage ever started becoming less than the Vbc+Vce, the base current would start going lower, which would start increasing the collector-emitter channel resistance, so this sort of a negative feedback prevents the transistor from ever being so saturated that it drops less voltage via the combination of BC and CE voltages than across the BE alone.
EDIT: Your question in the comment below has helped me figure out a better answer.
Let's start with an unsaturated transistor working in its linear region, as below.
The base-emitter junction is forward-biased with its typical forward diode voltage drop.
The base-collector junction is reverse biased.
Notice one thing here:
the Vbe voltage or the voltage across the base-emitter diode is equal to the sum of the Vbc and Vce voltages, in other words to the sum of the base-collector diode and collector-emitter channel voltage drops!
First an actual transistor schematic:
simulate this circuit
Now a diode representation "equivalent":
simulate this circuit
Now we look at the saturated transistor situation:
Again, first the transistor schematic:
simulate this circuit
And now the diode "equivalent":
simulate this circuit
Notice that we have exactly the same thing here with a saturated transistor as we did above with the non-saturated transistor:
the Vbe voltage or the voltage across the base-emitter diode is equal to the sum of the Vbc and Vce voltages, in other words to the sum of the base-collector diode and collector-emitter channel voltage drops!
So, apparently a bipolar junction transistor becomes saturated when its base-collector junction becomes forward biased and its voltage equals the base-emitter junction voltage.
Additionally, driving such a transistor deeper into saturation allows for a lower voltage drop across the collector-emitter channel, and this voltage adds to the base-collector diode voltage drop to equal the base-emitter voltage drop.
One more thing to notice is that the collector-emitter channel in all cases acts like a variable resistor whose resistance depends on the amount of the base current being injected into a part of the channel.
Just for an exercise and a further clarification, let's make the values of resistors such that the base-emitter diode and collector-emitter channel voltages are equal.
This is the point at which a transistor starts to enter into a saturation.
Notice that the voltage between base and collector or on the base-collector diode is actually ZERO!
What does this mean?
Does it mean that the base-collector diode/junction is shorted?
NO! Such thing can't happen inside a functional transistor, although it would seem to be the case based on zero voltage between base and collector.
What is ACTUALLY happening here is that the base current value has changed the collector-emitter resistance to the value which in that particular circuit (in combination with a particular load) develops a voltage drop equal to the base-emitter voltage at that moment.
The base and the collector electrodes are not shorted in this case, even if zero voltage between would make you believe they are.
This is another lesson to "take home": zero volts between two points or equal potential on two different points does not automatically mean they are shorted or connected to each other; it may mean that they are simply at the same potential at the moment.
simulate this circuit
Again, a diode "equivalent" schematic which should make what's going on a little more obvious:
simulate this circuit
OLD ANSWER BELOW:
No, the Ic does NOT reverse its direction. It still keeps flowing from collector to emitter (assuming positive/conventional current flow here).
simulate this circuit
As you can see in the above circuits, the one on the left is unsaturated, and the collector potential is above the base potential.
The one on the right is saturated, and the collector potential is lower than the base potential.
In both cases, the emitter current is equal to the sum of the base and collector currents.
Feel free to click on "simulate this circuit" to see it better and experiment with it.
And yes, the saturation happens when Vbe is larger than Vce, as the Vbe can't go below a certain voltage (typically around 0.6V for silicon transistor) if you want to keep the transistor switched ON, while the Vce saturation voltage can go close to 0V.
In that case, both PN junctions are forward biased.