Update: I think it will be useful to have a more coherently written answer, since I have learned more since my original response. Under the assumptions of the question, the answer is yes, the vertex operator algebra $V$ will be regular, provided that the categorical dimension $\mathrm{dim}_{\mathcal{C}}\,V\neq 0$, where $\mathcal{C}$ is the modular tensor category of $U$-modules. So for instance if $U$ is a unitary regular VOA and $V$ is a unitary $U$-module, then the categorical dimension of $V$ will be a positive real number and $V$ will be regular. I'm not aware of any pathological extensions of non-unitary regular VOAs that have categorical dimension $0$, but I also don't know that these can be ruled out.
As noted in the question, to show that $V$ is regular, it is enough to show that $V$ is rational, i.e., show that if $W=\bigoplus_{n\geq 0} W(n)$ is an $\mathbb{N}$-gradable weak $V$-module, then $W$ is the direct sum of irreducible ordinary $V$-modules (modules $W$ for which the $W(n)$ are finite dimensional). The rationality of $V$ follows from three facts:
$V$ is $C_2$-cofinite. This holds because $U$, as a regular VOA of CFT type, is $C_2$-cofinite, and therefore all ordinary $U$-modules, including $V$, are $C_2$-cofinite.
The category of ordinary $V$-modules is semisimple. To prove this, one uses the results from https://arxiv.org/abs/1406.3420 that $V$ is a commutative associative algebra in the modular tensor category $\mathcal{C}$ and that the category of ordinary $V$-modules is the category of "dyslectic" modules $\mathrm{Rep}^0\,V$ for the algebra object $V$. Then Kirillov and Ostrik showed in https://arxiv.org/abs/math/0101219 that $\mathrm{Rep}^0\,V$ is semisimple under the assumption $\mathrm{dim}_{\mathcal{C}}\,V\neq 0$. (I am not sure how necessary the assumption on non-vanishing of the dimension is in the VOA context, but it is certainly necessary in other contexts: recall the converse to Maschke's Theorem.)
- If the ordinary module category of a $C_2$-cofinite VOA $V$ is semisimple, then $V$ is rational. This was proved by Carnahan and Miyamoto in Lemma 3.6 and Proposition 3.7 of https://arxiv.org/abs/1603.05645.
Another proof of 3. goes as follows. Let $W=\bigoplus_{n\geq 0} W(n)$ be an $\mathbb{N}$-gradable weak $V$-module; we need to show that $W$ is a direct sum of irreducible ordinary $V$-modules.
First assume that $W$ is generated by a homogeneous vector $w\in W(N)$ for some $N\geq 0$. Then by standard vertex algebra theory,
$$ W = \mathrm{span}\lbrace v_n w\,\vert\,v\in V, n\in\mathbb{Z}\rbrace, $$
and hence
$$ W(N)=\mathrm{span}\lbrace v_{\mathrm{wt}\,v-1} w\,\vert\,v\,\mathrm{homogeneous}\rbrace. $$
This means that $W(N)$ is singly-generated as a module for the $N$th Zhu's algebra $A_N(V)$ as in https://arxiv.org/abs/q-alg/9612010. But since $V$ is $C_2$-cofinite, $V$ is also $C_{2N+2}$-cofinite and therefore $A_N(V)$ is finite dimensional by Proposition 2.14 in https://arxiv.org/abs/0712.4109 (where the result is attributed to Miyamoto). Hence $W(N)$ is finite dimensional and generates $W$, and then Buhl's results in https://arxiv.org/abs/math/0111296 on spanning sets of weak modules for $C_2$-cofinite VOAs imply that $W$ is an ordinary $V$-module (see his Corollary 5.6). Thus $W$ is semisimple because the category of ordinary $V$-modules is semisimple.
Now take a general $\mathbb{N}$-gradable weak $W$. Then $W=\sum_{n\geq 0}\sum_{w\in W(n)} V\cdot w,$ where $V\cdot w$ is the $\mathbb{N}$-gradable weak $V$-submodule of $W$ generated by homogeneous $w$. By the previous case, each $V\cdot w$ is an ordinary and thus semisimple $V$-module. This means that $W$ is a sum, and therefore also a direct sum, of ordinary simple $V$-modules, completing the proof that $V$ is rational.
Original answer: 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:
$V$ is a simple VOA, and
$\mathrm{dim}_{\mathcal{C}}\,V\neq 0$.
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.
