Cat urine penetrates deep, dries and crystalizes. Nothing will break it down except very harsh specific chemicals. And even then, you may be out of luck.
We bought a condo with a cat urine problem. I found a guy who inspected with black light and sold me a "non-over the counter" chemical product.
We tore up the carpets, ripped up the pergo flooring, chemically treated the hard wood floor and then sanded and sealed them. In the basement before putting new carpeting down again, we rolled on some kind of barrier paint special for keeping moisture down. Running a dehumidifier will help too. Also, in our case, it was a female cat that lived there. So, we did not have to worry as much about the walls. Otherwise, you may have to rip out baseboards or drywall too. It was a lot of work but results were good.
I recommend you really do your homework on finding a good chemical product. Most over the counter products can not actually break down that crystal. I really don't think you'll be able to get rid of the smell with any natural type of product. Sorry.
One other thing you may want to look into because of the allergies is if there is mold in the basement. In our current home, we hired someone to take an air sample and results came back within a week.