The answer to your question is No, using versatile weapons with two hands does not disqualify them as monk weapons.
But not because of any reason that made you ask the question, because your whole question is wrong as it's clear from your question you haven't understood the paragraph in question.
The paragraph in question on pg. 78, PHB says (emphasis mine):
At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are shortswords and any simple melee weapons that don't have the two-handed or heavy property.
Let's look at what that paragraph really says.
The first thing it does is stating that at 1st level a monk has a master of different combat styles using unarmed strikes and monk weapons.
The second thing it does is stating that monk weapons are shortswords and any simple melee weapon that doesn't have either the heavy property or the two-handed property.
That's everything that paragraph states. It simply tells you that monks fight unarmed or with monk weapons and then defines monk weapons as either shortswords or any simple melee weapon that isn't heavy or requires two-hands to use.
Simple melee weapon with heavy or two-handed property = not monk weapons.
Simple melee weapon without heavy or two-handed property = monk weapons.
So what about the versatile property? A weapon with that property just gives you the option to use two hands during an attack for greater damage.
However, in this case whether a weapon has the versatile property or not has no impact on the RAW as it only purpose in this case is to define what a monk weapon is, which is a shortsword or any simple melee weapon without either the heavy property or the two-handed property. Any other properties such as light, finesse, reach or versatile has in this case no bearing on the rules. Nor does how you use said weapon.
What do they convert into?
The Weapons and Size section on page 220 of the Player's Handbook says that:
Large, Huge, and Gargantuan creatures use weapons that are specifically sized for them. Each size category larger than Medium increases the weapon's damage die by one size.
This is followed by tables indicating the damage die size progression for each of one- and two-handed weapons:
1d4 -> 1d6 -> 1d8 -> 1d10 -> 1d12 -> 2d6 -> 2d8 -> 2d10
1d8 -> 2d4 -> 1d10 -> 1d12 -> 2d6 -> 2d8 -> 2d10
So, since a medium longbow (which is two-handed) does 1d10 damage, a large one does 1d12. This increases the average damage from 5.5 to 6.5. Similarly, since a medium greatbow (which is two-handed) does 1d12, a large one does 2d6. This increases the average damage from 6.5 to 7.
If I can use weapons one size larger, can I use this to have two two-handed weapons in my original size?
Bugbears and Oversized are defined in the Monster Manual on page 276.
Bugbears are medium size. Oversized says that:
You can use weapons of your size or one size larger than you as if they were your size.
Since you can already use medium weapons as if they're medium, all this says is that you can use large weapons as if they were medium. It does not make you large, nor does it change the size of the large weapons, nor does it let you use weapons of any other size as if they were any other size, i.e. you can still only use small weapons as if they were small, not as if they were tiny.
Therefore, no, you cannot wield two medium two-handed weapons. You still treat them as if they were medium.
If number 2 works, does it work with longbow/greatbow or any other ranged weapon?
We've already established that number 2 doesn't work.
In the previously mentioned Weapons and Size section, it also states that:
Large creatures can use two-handed weapons intended for creatures one size category smaller than themselves and treat them as one-handed weapons. A fire giant (Large) can use a human's greatsword with one hand, and a fire titan (Huge) can use a fire giant's greatsword with one hand. A creature can't use an undersized one-handed weapon at all; its hand is too large to effectively hold the weapon's small grip.
Although it's not written explicitly, the examples make it clear that this rule is intended to apply to all creatures, not just Large ones, i.e.:
Creatures can use two-handed weapons intended for creatures one size category smaller than themselves and treat them as one-handed weapons.
In this case, it appears that you can wield a small two-handed ranged weapon, such as a small longbow, in each hand. This doesn't make much sense, and I would expect a good DM to rule against it on those grounds, but technically I think you could fire the longbows while wielding one in each hand.
That being said, you couldn't reload either of them.
On page 217 of the Player's Handbook, under Load:, it says that:
Any weapon that has the load property requires two hands to load, even if you can use only one hand to attack with it. (The sling, for example, is a one-handed weapon, but you need a free hand to load it.)
All ranged weapons that use ammunition have the Load property, including all bows and crossbows.
While Netzach makes a very good point, I'd like to add in the die dynamics.
Although the average is very similar the two situations behave very differently.
Rolling 2 dice creates a bell curve distribution of the possible values. In this case the average (7) has a chance of 1 in 6, while the max has a chance of 1 in 36. Half of your rolls will be 6,7, or 8. With the addition of more dice this curve gets steeper. 4d6 would have an average about 14 (1 in 36) and a max of 24 (1 in 648)
All numbers have an equal chance of occurring (a uniform distribution), thus your chance of a 12 is 1 in 12. Only one quarter of your rolls will be a 6,7, or 8. And on a crit, 24 is 1 in 144. You have a better chance of rolling higher than average, but also a higher chance of rolling lower than average. Many would argue that this balances out. But when you are in a battle and need a 12, your chances are better with the axe.
Some prefer the Greatsword because the high chance of average damage is "slow but steady".
Others accept the risk of a low roll with the Great Axe in order to have a better chance at high damage.