Okay, without going into actual war tactics, siege tactics, Lord of the Rings lore, book-vs-movie discussion, and all that other fancy stuff that would turn this into a nightmare (but a fun nightmare) and focusing SOLELY on what was shown happening at Helm's Deep during the fight and before the riders showed up, here's my take:
Yes, the Uruks would have. Here is the evidence that I give to support this:
The elves, for all their archery glory and renowned fighting ability, were getting slaughtered. If Aragorn, Leagolas, and Gimili had not been on that wall, I'm certain the elves would have been overrun long before the bomb went off. I've watched that battle countless times, and every time I see the elves losing. Recall when that one beserker managed to get on the wall. He swung his blade left, then right, and took down about four or five elves in a matter of seconds. Also, if you look at the actors in the background, you will never see an elf winning a fight (as in, impaling an Uruk or kicking them off the ladder). They are either struggling to hold off the Uruk, or losing.
Several elves get pulled off the wall and thrown to their deaths by Uruks climbing the ladder. One of the scenes that has always stuck with me is a computer animated scene. An Uruk reaches up--from the ladder, mind you--grabs an elf by the ankle, and pulls him off the wall. Why was that elf standing so close to the edge? It's different for the poor elf who got picked up and tossed over the wall, but this guy was begging for it. Not exactly the smartest choice if you ask me.
When Haldir dies, he looks around before he falls. He sees a lot more bodies of his people than he does of the Uruks. The elves are piled along the wall, and it's meant to show that Haldir fears they have lost.
There are NO elves when the war is done. Okay, there's one, and his final count was forty-two, but he doesn't count. I'm sure some elves survived, but they were so few in number that we didn't see them at the end of the battle. The Uruks basically wiped them out. And elves are better warriors than farmers.
The Uruks breach the gate. Again. Remember, Gimili and Aragorn have to go outside so the King has time to shore up the gate. However, if you look at the gate while Gamdling (is that how you spell it?) is yelling for everyone to retreat, we see the gate is smashed open again. This has nothing to do with the retreat. The gate just happened to give while they were retreating. Meaning it would have given even if they had stood and fought. Considering that the Uruks can now storm the courtyard, instead of coming up ladders one at a time, it would have been a slaughter.
There was a way out of the caves, but it would have done little good. The Uruks would have smashed down the door to the caves eventually. Once inside, they would have overrun the remaining defenders and gone after the people who had escaped. As we saw, Uruks can run for an INSANE distance without rest. They would have easily chased down women and children and wounded soldiers and ended them.
So no. If Gandalf, the trees (it wasn't the Ents, remember, but the trees themselves), or Eomer hadn't shown up and Helm's Deep was left on its own, it would have fallen and the White Hand would have claimed victory.