As far as modularity of the category of $V$-modules is concerned, $V$ will effectively be regular under suitable conditions. The point is that since $V$ will be a commutative associative algebra object in the modular tensor category of $U$-modules, one can study its representations using the techniques developed in, for instance, Kirillov and Ostrik's paper https://arxiv.org/abs/math/0101219. They proved that if the base category $\mathcal{C}$ (in this case of $U$-modules) is modular, then so is the category of "dyslectic" modules for the extension provided that:

1. $V$ is a simple VOA, and

2. $\mathrm{dim}_{\mathcal{C}}\,V\neq 0$.

So for instance if $U$ is a unitary regular VOA, this result will apply because all quantum dimensions will be positive real numbers. I'm not aware of any pathological extensions of non-unitary regular VOAs that have quantum dimension $0$, but I also don't know that these can be ruled out.

I should also mention that to apply the tensor-categorical work of Kirillov and Ostrik to VOAs, one also needs the results in https://arxiv.org/abs/1406.3420 and my paper https://arxiv.org/abs/1705.05017 with Thomas Creutzig and Shashank Kanade showing that the tensor category of dyslectic modules for the extended algebra studied by Kirillov and Ostrik actually agrees with the vertex algebraic tensor category structure on the category of ordinary $V$-modules.

As far as whether the extended algebra $V$ is regular in the original sense of Dong, Li, and Mason, the above results certainly suggest that the answer should be yes, at least as long as $\mathrm{dim}_\mathcal{C}\,V\neq 0$, but I am not aware of whether this is known in general. One case that has been known for some time is the case of framed VOAs, i.e., conformal extensions of a tensor power of $c=\frac{1}{2}$ simple (and unitary!) Virasoro VOAs; the regularity of framed VOAs was proven in Theorem 2.12 and Corollary 2.13 of https://arxiv.org/abs/q-alg/9707008.