No.
As written, the War domain power is:
Free Martial Weapon Proficiency with deity’s favored weapon (if necessary) and Weapon Focus with the deity’s favored weapon.
(Pulled from d20SRD.org, and I have checked that it is a verbatim match for the 2004 printing of the 3.5 PHB).
If you don't have a deity, your non-existent deity doesn't have a favored weapon. You might be able to argue that this still fulfills pre-reqs, if you could find a pre-req which was "have any martial weapon proficiency," but even that would be somewhat iffy.
And really, there's not a lot of need to worry about whether Clerics are getting enough toys, especially Clerics of causes.
I remember seeing a chart showing weapons that could be used as other weapons of different sizes. E.G. a medium shortsword is equal to a small longsword. Both do 1d6 damage and have a critical rating of x2 on 19-20. This would allow a small character to use a medium shortsword as a small longsword. (I think this is not RAW)
This sounds like it might have been referencing the 3.0 rules, which did something kind of like that. As you are playing 3.5, best to ignore it. A mis-sized weapon remains that type of weapon, just wielded differently.
Anyway, on to the meat of the question: yes, if you have proficiency in it, the dwarven waraxe can be a one-handed weapon. If it is one size too large for you, it becomes a two-handed weapon for you, and you take a −2 penalty on attack rolls with it. So at best, assuming you are a dwarf, you can take a −2 attack penalty to go from a 1d10 (average 5½) damage weapon to a 2d8 (average 9) weapon. If you are not a dwarf, this also costs a feat, or the penalty balloons to an untenable −6.
Trading −2 attack for +3½ damage isn’t really a great trade, seeing as Power Attack would get you +4 for the same −2 penalty and can go beyond that. Realistically, you don’t want to take any attack penalties you can avoid—Power Attack loves Shock Trooper because it eliminates the attack penalty, and likewise, anyone wielding an oversized weapon wants a way to do so without penalty.
There is a terrible feat for that, Monkey Grip, which halves the penalty. A feat is a deeply precious thing, far too great a cost to only halve the penalty. Far superior to that are powerful build or the strongarm bracers from Magic Item Compendium. The best ways to get powerful build is the goliath race from Races of Stone, but it has LA +1 and isn’t really worth it (though the goliath barbarian substitution levels in Races of Stone can tilt the needle in their favor). Meanwhile the strongarm bracers cost just 6,000 gp. Since you can only have one race, and strongarm bracers explicitly do not stack with powerful build, the bracers are the way to go. Spending 6,000 gp on a +2 bonus to attack is clearly worth it.
But then there is no point in using a dwarven waraxe: with strongarm bracers, you can wield weapons as if you were a size category larger to begin with. That means you can just use a large two-handed weapon: a large, 3d6 (average 10½) damage greataxe represents a +5 damage bonus instead of +3½.
You could also use the strongarm bracers to wield a huge dwarven waraxe as if you were large, i.e. with a −2 penalty. That would be 3d8 (average 13½) damage, another 3½ greater than the large greataxe. Again, though, −2 attack for +3½ damage isn’t a great trade, and since you’re already using the strongarm bracers, easy approaches to dealing with that penalty aren’t available. You could go with the Wield Oversized Weapon feat, but again, a feat is a huge cost—and now this needs two of them (since you need to have Monkey Grip). And even if you did, again you would want to go with the 4d6 (average 14) damage huge greataxe over the huge dwarven waraxe. Two feats for +4 damage is pretty awful, though.
But we can actually go even harder with this. You might go with a gargantuan greataxe, for 6d6 (average 21) damage, since Wield Oversize Weapon makes it simultaneously count as a size category smaller and a one tick “lighter” in effort, i.e. you can treat it as a huge one-handed weapon (that you can wield as a large two-handed weapon at a −2 penalty). Two feats for −2 attack and a +10½ damage is starting to look kind of good, though you definitely can do better.
In short: it seems to me that the best approach here is to use a pair of strongarm bracers, and to use a large greataxe, rather than a large dwarven waraxe. That, or if you really felt like you could take the penalty and wanted to squeeze the most out of this, go with Monkey Grip, Wield Oversized Weapon, and strongarm bracers to wield a gargantuan greataxe at a −2 penalty.
You’d still do vastly more damage with a lance, though, and abusing charge damage multipliers.
Best Answer
Outside of Complete Scoundrel (14) and Sharn: City of Towers (168), the prestige class exotic weapon master is unmentioned beyond its original appearance in Masters of the Wild (53–4) then its revised appearance in Complete Warrior (30–1). Neither Scoundrel nor Sharn offers anymore insight than Warrior into how to use the exotic weapon master prestige class's extraordinary ability exotic weapon stunt (uncanny blow). That means you're stuck reading the special ability and making a choice.
What uncanny blow says…
The extraordinary ability exotic weapon stunt (uncanny blow) says
(If that last sentence strikes you as a largely unnecessary reminder, it should. This reader suspects that the precise wording of the 3.5 revision's of the feat Power Attack (PH 98) was still being decided when Complete Warrior was published, Warrior appearing on shelves only 5 months after the revised Player's Handbook.)
Anyway, the Player's Handbook description of the bastard sword, in part, says, "A bastard sword… is an exotic weapon. A character can use a bastard sword two-handed as a martial weapon" (121). And the bastard sword appears on Table 7–5: Weapons under the heading Exotic Weapons under the subheading One-handed Weapons (117). For this reader, that's enough: the bastard sword is always a one-handed exotic weapon that can—it just so happens and if the wielder so chooses—also be used as a martial weapon if the wielder employs it two-handed.
Thus this GM would rule—perhaps even a little reluctantly, depending on the optimization level of the campaign and the possible outcome of such a ruling—that a creature that's proficient with all martial weapons, that enters the prestige class exotic weapon master, and that picks the special ability exotic weapon stunt (uncanny blow) can use that special ability with a bastard sword that the creature wields in two hands, despite the creature not possessing the feat Exotic Weapon Proficiency (bastard sword) (PH 94) yet possessing the feat Weapon Focus (bastard sword) (PH 102) (perhaps through a workaround like the swordsage's extraordinary ability discipline focus (Weapon Focus (Diamond Mind)) (Tome of Battle 16).
…Then there's some Sage Advice
The Dragon #316 Wizards Workshop column "Sage Advice: Equipment and Combat Official Answers" (100–4) includes two exchanges relevant to the extraordinary ability exotic weapon stunt (uncanny blow).
So, according to this advice, a creature that's wielding a bastard sword in two hands is using that bastard sword as a two-handed weapon. That means the special ability exotic weapon stunt (uncanny blow) can apply to the bastard sword, and we're good to go, right? Not so fast. The very next question is this one:
(Emphasis in answers mine. The Sage at the time is Monster Manual primary author and dnd-3e co-designer Skip Williams.) So, in sum, the advice tells the reader that Table 7–5: Weapons lists the bastard sword and dwarven waraxe as one-handed weapons only for convenience and that outside of combat, those weapons are totally two-handed weapons, even though in combat those weapons can be considered either one-handed or two-handed weapons depending on how they're used.
So an argument can be made either way. On one hand, the description of the stunt uncanny blow could be describing how a creature must be using the exotic weapon in combat, so the stunt totally falls under the first ruling, making using the stunt with a bastard sword that's used two-handed legit. On the other hand, the description of the stunt could be describing in the abstract the kind of weapon that the creature must wield to experience the stunt's bonuses, and that description is independent of the weapon's wielder, in which case the stunt's not legit for use with the bastard sword or dwarven waraxe because, technically, no matter how they're used those weapons are always two-handed weapons.
This reader tends to favor the former rather than the latter: the universe is hard enough on the mundane warrior without—as it seems to this reader—double-secret rules prohibiting players from their PCs being ever-so-slightly improved and more versatile axeman or swordsman.
Note: These Sage Advice exchanges are repeated nearly verbatim by the Main FAQ on pages 48 and 57, respectively. Issues with the FAQ can be viewed in answer to this question.