Persistence time of ozone
Let's first consider the scenario in the absence of free radicals that can act as catalysts of $O_3$ destruction.
In such scenario, added to the absence of UV radiation, the photolysis of ozone ($O_3 + UV → O_2 + O$) would not be possible. Therefore, the only way to destroy ozone would be by the reaction

$O_3 + O· → 2 O_2$

But the lack of UV radiation would also stop the production of atomic oxygen by photolysis, therefore only the preexisting atomic oxygen would be available for that reaction.
Then, for simplicity we can consider two extreme scenarios:


All the atomic Oxygen forms $O_2$ and the Ozone level remains constant at the same level it has prior to the shut down of UV radiation.
All the atomic Oxygen is used to destroy Ozone


In the second scenario the final concentration of $O_3$ will depend on the initial concentration of it and atomic oxygen. But as seen in the following image, the stratospheric concentration of atomic oxygen is much lower than the one of $O_3$, therefore the ozone levels would never fall to zero.


(Figure taken from here)

Let's now consider the more realistic case, on which there are other "polutants" that can break down ozone. The moste common ones are hydroxyl radical (OH·), nitric oxide radical (NO·), chlorine radical (Cl·) and bromine radical (Br·) (source), and the first two are the only ones naturally occurring. Therefore, the ozone remaining from our first scenario would be reduced to zero by reactions like:

$OH + O_3 → {HO}_2 + O_2$

or

${NO}_2 + O_3 → {NO}_3 + O_2$

Among others.

The exact time that will take will depend on the elevation, as the abundance of ozone as well as the radicals can vary orders of magnitude trough the stratosphere. It will also depend on the exact spectrum of your "sustitute Sun", as some of those reactions are also mediated by light and might not happen in the absence of light of the right wavelength.

I know this answer might be disappointing, as it seem that a model would need to be run to really say how long it would take to destroy all stratospheric ozone. But still I thought would be an informative answer after 11 month without one. I hope, it motivate a better answer or the improvement of this one.