There is a very general criterion for a map on $\pi_1$ to be surjective. Recall that for $X$ connected, the category of finite étale covers of $X$ is equivalent to the category $\pi_1(X)\text{ -}\operatorname{Set}_f$ of finite sets with a continuous $\pi_1(X)$-action. Under this correspondence, the $Y \to X$ finite étale with $Y$ connected correspond to the connected $\pi_1(X)$-sets $S$ (i.e. $\pi_1(X)$ acts transitively on $S$).
Lemma. Assume $X$, $Y$ connected, and $f \colon X \to Y$ a morphism. Then the induced morphism $\pi_1(f) \colon \pi_1(X) \to \pi_1(Y)$ is surjective if and only if for every $Z \to Y$ finite étale with $Z$ connected, the pullback $Z_X \to X$ is connected.
Proof. If $\pi_1(f)$ is surjective, then clearly any connected $\pi_1(Y)$-set is connected as $\pi_1(X)$-set. Conversely, if $\pi_1(f)$ is not surjective, then some $\gamma \in \pi_1(Y)$ is not in the image. Since fundamental groups are profinite, the image of $\pi_1(f)$ is closed, so the image of $\pi_1(f)$ misses some open neighbourhood of $\gamma$. Thus, there exists an open subgroup $U \subseteq \pi_1(Y)$ such that
$$\gamma U \cap \operatorname{im} \pi_1(f) = \varnothing.$$
Then the finite $\pi_1(Y)$-set $S = \pi_1(Y)/U$ is not connected as $\pi_1(X)$-set. But it is clearly connected as $\pi_1(Y)$-set. $\square$
To apply this to the specific geometric setting you are interested in, just note that if $f \colon X \to Y$ has connected geometric fibres, then the same holds for the base change to any finite étale covering $Z \to Y$. It is then clear that if $Z$ is connected, so is $Z \times_Y X$.
Remark. There are more equivalent criteria for surjectivity; see for example Tag 0B6N. The one I gave above is amongst the ones listed, but this was not the case at the time of writing; hence my writing out the proof. My proof above was originally part of the proof of Tag 0BTX.
This is an expansion of my comments above. You do not need resolution of singularities or SGA 4. The key step is "elimination of
ramification" or "Abhyankar's Lemma". This is proved in Append. 1 of Exposé XIII of SGA 1. Here is the link in the Stacks
Project.
Abhankar's Lemma, Stacks Project Tag 0BRM
http://stacks.math.columbia.edu/tag/0BRM
Here is the setup for Abhyankar's Lemma. Let $A$ be a
DVR that contains a characteristic $0$ field, let $A\subset B$ be an
injective, local homomorphism of DVRs with ramification index $e$, i.e.,
$\mathfrak{m}_B^{e+1}\subset \mathfrak{m}_AB \subset \mathfrak{m}_B^e$, let
$K_1/\text{Frac}(A)$ be a finite field extension, let $L_1/\text{Frac}(B)$
be a compositum of $K_1/\text{Frac}(A)$ and
$\text{Frac}(B)/\text{Frac}(A)$, let $A\subset A_1$, resp. $B\subset B_1$,
be the integral closure of $A$ in $K_1$, resp. of $B$ in $L_1$. Let
$\mathfrak{n}_A\subset A_1$ be a maximal ideal that contains $\mathfrak{m}_A
A_1$, and let $\mathfrak{n}_B\subset B_1$ be any maximal ideal that contains
$\mathfrak{m}_B B_1 + \mathfrak{n}_A B_1$.
Abhyankar's Lemma.
if the ramification index $e$ of $A\subset B$ divides the ramification index
of $A\to (A_1)_{\mathfrak{n}_A}$, then $(A_1)_{\mathfrak{n}_A}\subset
(B_1)_{\mathfrak{n}_B}$ is formally smooth, i.e., the ramification index
equals $1$.
Nota bene. In characteristic $0$ this follows easily from the Cohen
Structure Theorem, etc. In positive characteristic, Abhyankar's lemma says
more, because (1) the tensor product $\text{Frac}(B)\otimes_{\text{Frac}(A)}
K_1$ may be nonreduced, and (2) under the assumption that the the residue
field extension $A/\mathfrak{m}_A \to B/\mathfrak{m}_B$ is separable, we
need to also prove that the residue field extension of
$(A_1)_{\mathfrak{n}_A} \to (B_1)_{\mathfrak{n}_B}$ is separable. This
requires a further assumption that $e$ is prime to $p$. When $e$ is not
prime to $p$, Abhyankar's Lemma has counterexamples, but the result that
sometimes does the job is Krasner's Lemma, Stacks Project Tag 0BU9:
http://stacks.math.columbia.edu/tag/0BU9
Let $K$ be a field (not necessarily characteristic $0$), let
$X_K$ be a projective, connected $K$-scheme, and let $x_0\in X_K$ be a $K$-rational point. For every $K$-scheme $T$, an étale cover of $X_T$ trivialized over $x_0$ is a finite, étale morphism $g_T:Z_T\to X_T$ of some degree $d$ together with an ordered $n$-tuple of $T$-morphisms $(s_i:T\to Z_T)_{i=1,\dots,d}$ such that the union of the images of the sections $s_i$ equals $Z_T\times_{X_K} \text{Spec}\kappa(x_0)$.
Rigidity in the Projective Case. For every $K$-scheme $T$, every étale cover of $X_T$ trivialized over $x_0$ is isomorphic to the base change of an étale cover of $X_K$ trivialized over $x_0$, and that trivialized étale cover is unique up to unique isomorphism.
This is, essentially, proved in Section 1 of Exposé X of SGA 1. The key point is rigidity of trivialized étale covers in the proper case.
Descent for Affine Curves.
Now assume further that $K$ is a characteristic $0$ field that contains all roots of unity. Assume that $X_K$ is a smooth, projective, connected curve over $K$. Let
$Y_K\subset X_K$ be a
proper closed subset, i.e., a finite set of closed points. Denote
$X_K\setminus Y_K$ by $U_K$, and assume that the $K$-point $x_0$ is contained in $U_K$.
Fix an integer $d$.
Let
$f_K:\widetilde{X}_K\to X_K$ be a finite surjective morphism of some degree
$n$ with $\widetilde{X}_K$ a smooth, projective curve such that (i)
$f_K^{*}(x_0)$ is a set of $n$ distinct $K$-rational points of
$\widetilde{X}_K,$ and such that for every closed point $y\in Y_K,$ for
every closed point $\widetilde{y}\in \widetilde{X}_K$ with
$f(\widetilde{y})=y$, the ramification index of $\mathcal{O}_{X_K,y}\to
\mathcal{O}_{\widetilde{X}_K,\widetilde{y}}$ is divisible by $e$ for every
$e\leq d$. For instance, begin with a finite morphism $g:X_K\to
\mathbb{P}^1_K$ that is smooth at every point of $Y_K$, such that $g(x_0)$
equals $[1,1]$, and such that $Y_K\subset
f^{-1}(\underline{0}+\underline{\infty})$, and define $\widetilde{X}_K$ to
be the normalization of the fiber product of $g$ and the morphism
$\mathbb{P}^1_K\to \mathbb{P}^1_K$ by $[z_0,z_1]\mapsto
[z_0^{d!},z_1^{d!}]$.
For a field extension $L/K$, the ramification hypothesis on
$f_L:\widetilde{X}_L\to X_L$ over $Y_L$ is still valid. For every finite
surjective morphism $W_L\to X_L$ of degree $d$, for every closed point $w$
of $W_L$ that maps to $Y_L$, the ramification index $e$ at $w$ divides $d!$.
Thus, by Abhyankar's Lemma, the normalization $\widetilde{W}_L$ of
$W_L\times_{X_L} \widetilde{X}_L$ is étale over $\widetilde{X}_L$ at
every closed point lying over $Y_L$. Thus, if $W_L\to X_L$ is assumed to be
étale away from $Y_L$ then $\widetilde{W}_L\to \widetilde{X}_L$ is
everywhere étale of finite degree $d$. Assume further that the fiber of $W_L$ over $x_0$ is a set of $d$ distinct $L$-rational points. Then also the fiber of $\widetilde{W}_L$ is a set of distinct $L$-rational points. By the projective descent result above, there exists a finite, étale morphism $\widetilde{W}_K\to
\widetilde{X}_K$ whose fiber over $x_0$ is a set of distinct $K$-rational points and whose base change equals $\widetilde{W}_L$. Thus, $W_L$ is
an intermediate extension of the base change to $L$ of the extension
$\widetilde{W}_K\to X_K$. In particular, for the fiber $\widetilde{W}_K \times_{X_K} \text{Spec}\kappa(x_0)$ over $x_0$, the degree $n$ morphism $\widetilde{W}_L \to W_L$ defines a partition $\Pi$ of the fiber into $d$ subsets of size $n$.
Descent for $W_L$. There exists a unique irreducible component $W_K$ of the relative Hilbert scheme $\text{Hilb}^n_{\widetilde{W}_K/X_K} \to X_K$ whose fiber over $x_0$ parameterizes the partition $\Pi$ above. The base change of $W_K\to X_K$ to $L$ is isomorphic to $W_L\to X_L$.
In conclusion, for the open
subset $U_K=X_K\setminus Y_K$, for every finite étale morphism
$V_L\to U_L$, there exists a finite étale morphism $V_K\to U_K$ whose
base change to $L$ equals $V_L\to U_L$.
Descent in Arbitrary Dimension.
Now let $k$ be a characteristic $0$ field containing all roots of unity, and let $U_k$ be a normal, quasi-projective variety of dimension $r\geq 1$ together with a specified $k$-rational point $u_0$ in the smooth locus. I claim that there exists a blowing up $\nu_k:U'_k\to U_k$ at finitely many points including $u_0$ such that $U'_k$ is normal, and there exists a flat morphism $\pi:U'_k\to \mathbb{P}^{r-1}_k$ such that the exceptionial divisor $E_0$ over $u_0$ is the image of a rational section of $\pi$. The easiest way to get this is to embed $U_k$ into a projective space $\mathbb{P}^{r+s}_k$, choose a linear $\mathbb{P}^s_k\subset \mathbb{P}^{r+s}_k$ that intersects $U'_k$ transversally in finitely many points including $u_0$, and then let $\pi$ be the restriction to $U_k$ of the linear projection away from $\mathbb{P}^s_k$. Now let $K$ be the fraction field $k(\mathbb{P}^{r-1}_k)$ of $\mathbb{P}^{r-1}_k$, and let $U_K$ be the generic fiber of $\pi$. Let $x_0$ be the $K$-rational point corresponding to the exceptional divisor $E_0$.
For every field extension $\ell/k$, for every finite étale morphism $V_\ell\to U_\ell$ whose fiber over $u_0$ is a set of distinct $\ell$-rational points, the fiber product with $\nu_\ell:U'_\ell\to U_\ell$ is a finite étale morphism to $U'_\ell$ that is trivialized over $E_0$. Thus, setting $L=k(\mathbb{P}^{r-1}_\ell)$, we obtain a finite étale morphism $V_L\to U_L$ whose fiber over $x_0$ is a set of distinct $L$-rational points. Applying the curve case, this descends the generic fiber of $V_\ell\to U_\ell$ to a variety $V_K$ over $K=k(\mathbb{P}^{r-1}_k)$. Finally, we can construct $V_k\to U_k$ as the integral closure of $U_k$ in the fraction field of the $V_K$.
Best Answer
To simplify notation, let me write $U_i$ for $V\smallsetminus W_i$, and $U_{12}$ for $U_1\cap U_2=V\smallsetminus(W_1\cup W_2)$.
Fact: The obvious functor $$(\mathrm{Sch}/V)\longrightarrow (\mathrm{Sch}/U_1) \times_{(\mathrm{Sch}/U_{12})} (\mathrm{Sch}/U_2)$$ is an equivalence. In other words, a $V$-scheme $X$ is the same thing as a $U_1$-scheme $X_1$, a $U_2$-scheme $X_2$, and a $U_{12}$-isomorphism of their restrictions to $U_{12}$.
This is probably somewhere in EGA1. [EDIT: all I could find was section 2.4 of EGA1, relying on (4.1.7) of Chapter 0 (glueing of ringed spaces).]
However, we are dealing here with finite étale schemes, which happen to be affine over the base, so this boils down to the analogous statement for categories of quasicoherent sheaves, which is essentially trivial (plus the fact that ``finite étale'' is a local condition).
If we describe the categories of finite étale covers in terms of $\pi_1$-sets, the above equivalence says that the diagram of groups $$(*)\qquad\begin{array}{rcl} \pi_1(U_{12},p)=:G_{12}& \longrightarrow &G_1:=\pi_1(U_{1},p)\cr \downarrow && \downarrow\cr \pi_1(U_{2},p)=:G_2& \longrightarrow &G:=\pi_1(V,p) \end{array}$$ is cocartesian. In other words, we get the usual van Kampen statement: the natural map $$\pi_1(U_{1},p)\ast_{\pi_1(U_{12},p)}\pi_1(U_{2},p)\longrightarrow \pi_1(V,p)$$ is an isomorphism. [EDIT: the coproduct is in the profinite category, which perhaps makes it hard to describe in general. See Will Savin's comment.]
What we want to prove is that the map "on the other side" $$G_{12}\longrightarrow G_1 \times_G G_2$$ is surjective, given that all the maps in diagram ($\ast$) are surjective.
Identifying $G_i$ ($i=1,2$) with $G_{12}/N_i$, we see from the universal property of the coproduct that $G=G_{12}/N_{1}N_{2}$. [EDIT: clearly this works also in the profinite category: since $N_1$, $N_2$ are both compact normal subgroups, so is $N_1 N_2$, hence $G_{12}/N_{1}N_{2}$ is profinite].
Take any $(x_1,x_2)\in G_1 \times_G G_2$: thus we have $x_i=g_i N_i$ for some $g_i\in G_{12}$, and the fiber product condition says that $g_1=g_2 n_2 n_1$ for some $n_i\in N_i$ (recall that $N_1 N_2=N_2 N_1$). So, $(x_1,x_2)$ is the image of $g_1 n_{1}^{-1}=g_2 n_2\in G_{12}$. QED