As I Can Chan referenced, the entry for bastard sword states that it can be used as a two-handed martial weapon.
As we can see in the glossary of the PHB p.311
one-handed weapon: A weapon designed for use in one hand, such as a longsword, often either along with a shield or a light weapon in the other hand. A one-handed weapon is considered to be an object one size category smaller than its designated wielder (for example, a Medium longsword is a Small object)
a weapon designed for use in one hand...as are all the one-handed exotic weapons (such as the bastard sword that can also be used two-handed), is small. This also support the rule at p.113 of the PHB saying:
Weapon Size: Every weapon has a size category, such as Small,
Medium, or Large. This designation indicates the size of the creature for which the weapon was designed. A Small greatsword is a greatsword designed for a Small creature, such as a halfling.
Which, when we combine both rules, summarizes that a weapon is designed for a specific character size for its designed use, such as a medium bastard sword being designed for a medium character to use in one-hand.
We have to remember that exotic weapons is a weapon category as much as simple and martial and weapons in this category require a racial weapon proficiency or the exotic weapon proficiency feat. The bastard sword, and for example, the dwarven waraxe, have exceptions written in their description allowing these to be used as two-handed martial weapons though they are designed as one-handed exotic weapons.
As in any game, the DM has the final say on it but I would recommend that these kind of rulings be established before the campaign or character creation process as to ensure players can build their characters with proper expectations.
------------- Adding in response to wyrmwood as I cannot add this in comments---
The answer I gave is considering the Rules as Written. As all weapons have an entry in the Table 7.5 and all weapons have a description and there is no other place to look for a clear assignation of the "effort" it takes to use a weapon than the table itself, it should be used as the basis for it. Of course if there was an errata about this, it would take precedence but I haven't found any that states to change all exotic weapons one-handed similar to the bastard sword to set them as two-handed weapons.
Again, as we can see, the following supports the idea of using the table entry, it states that the weapon is designated to be used with such effort for a creature of its size (Weapon Size, p.113)
The measure of how much effort it takes to use a weapon
(whether the weapon is designated as a light, one-handed, or two handed weapon for a particular wielder) is altered by one step for each size category of difference between the wielder’s size and the size of the creature for which the weapon was designed.
Being designated as a one handed exotic weapon just means it requires a feat to use properly, like all other exotic weapons but is still one handed.
Now about the feats that change the weapon size, feats do not change the designated weapon size and I totally agree. An Exotic weapon requires the Exotic weapon proficiency feat to use and a Martial weapon requires the Martial weapon proficiency feat just as well so it is as hard to use a two handed sword for a wizard as it is to use a bastard sword in one or two hands.
You can of course determine that this is a medium sized object as the rule states that a one handed weapon is generally one size smaller than the wielder which means it is opened for exceptions but it will still be a one handed weapon. Also, the fact that the bastard sword description states (PHB. p.121)
Sword, Bastard: bastard swords are also known as hand-and-a-half swords. A bastard sword is too large to use in one hand without special training; thus, it is an exotic weapon. A character CAN use a bastard sword two-handed as a martial weapon. [bold and capital letters added by me for clarity]
means it is an option otherwise they could have written something like "The bastard sword is a two-handed martial weapon that can be used in one hand with proper training as an exotic weapon."
I think what is important here is, if you determine that a bastard sword is a medium object, then you can use this as any object size related rule (AC of worn or carried object for instance when attacking object) but it is still a one handed weapon. A weapon is an object but an object is not a weapon and the rule also say that it is the designated effort that determines the object size with a predetermined rule that applies "generally": two-handed weapons are same size as wielder, one-handed weapons are one size smaller than wielder and light weapons are two size smaller than wielder.
Changing the designated "effort" it takes to use a weapon, no matter the feat you put as required, have an impact in many combat options as these often refers to the weapon size and it's designated effort (light, one handed or two handed) such as Sunder. So you have to be careful in how you change the written rules but you are the master after all and if you decide to change the rules slightly, just make sure it is clear before starting your campaign to be fair with the players.