In Louisiana, if a widow dies with no children, how is property divided among surviving nieces/nephews and great nieces/nephews? If she dies intestate, will all property go only to a surviving niece; or is it also distributed to her great nieces and nephews.

You don't mention if any of her siblings are alive. Her nieces and nephews inherit if none are.
I am not sure how to read the law regarding grand nieces and nephews, when the niece or nephew predeceased her. Here is the link for those who are sure they understand.

Louisiana State Legislature. Select “Civil Code” and enter “880” in the “Article” box, if you want to read the code itself.

Intestate Succession in Louisiana | Nolo.com

If she had parents and/or siblings survive her then they would be in line before any nieces or nephews of any generation. Since non were mentioned, I'm going to guess that they are no longer living. In that case, I believe, the assets (after debts are paid of course) would still be divided between siblings equally, then each siblings share would be divided between his/her children equally. If by some tragedy one of her siblings only had grandchildren and no children left then those grandchildren or "great nieces/nephews" would stand to inherit their grandparents share.

'Divided between siblings equally, then each siblings share would be divided between his/her children equally.' Are you saying that deceased nieces and nephews count equally with the one remaining niece? That's not clear to me as I read the law.

From what I remember yes (provided each sibling had the same number of children). Say she had two brothers Sam and Bill they would each get 50% (provided parents are already deceased). But both have passed so children of Sam get 50% divided between them and children of Bill get 50% divided between them. So if Sam and Bill each had two children then each would get 25%. But if Sam had two and Bill had three it would not be equal fifths, Sam's children would each get half of 50% and each of Bill's would get one third of 50%.

I'm not in LA so it has been a while since I've had to go through their laws (Louisiana law specifically, because they don't tend to follow the norm of other states). But from what I remember that is how inheritance broke down. I've been trying to find a good link to confirm this (or correct it) but so far siblings and parents inheriting in the absence of spouse and children is as far as I have gotten on a reliable site.

http://goea.louisiana.gov/assets/leg...succession.pdf

This is about at clear as I can find. This still follows what I have already explained... but with bigger words.

We are talking about siblings of the one obvious HEIR, not the deceased. So I don't see how the Sam and Bill characters can each get 50%. That assumes she gets nothing.
But I agree that it looks like the children of the heir's siblings get a share. It just isn't clear in the link I provided to the actual code.