The two previous answers to the question show that $\phi$ is an isomorphism provided one the following conditions is satisfied:
(1) $(Y,y_0)$ is well-pointed.
(2) $Y$ is both locally simply connected at $y_0$ and first-countable at $y_0$.
I was not aware of Griffiths's theorem based on (2), but let me discuss the relationship between (1) and (2).
First note that (1) and (2) do not impose any restriction on $(X,x_0)$ so that we get a sort of "asymmetric" theorem. In contrast the Proposition in the question is symmetric in $(X,x_0)$ and $(Y,y_0)$.
Observation 1.
If $(Y,y_0)$ is well-pointed, then $Y$ is locally pointed contractible at $y_0$ and in particular locally simply connected at $y_0$.
From Lemma 4 in Strøm's paper it is known that $(Y,y_0)$ is well-pointed if and only if there exists a map $\varphi : Y \to I$ mit $y_0 \in \varphi^{-1}(0)$ and a homotopy $H : Y \times I \to Y$ such that
- $H(y,0) = y$ for all $y \in Y$.
- $H(y_0,t) = y_0$ for all $t \in I$.
- $H(y,t) = y_0$ for all $(y,t) \in Y \times I$ with $t > \varphi(y)$.
Now define $\Omega = \varphi^{-1}([0,1))$. This is an open neighborhood of $y_0$ in $Y$. The restriction $H' = H \mid_{\Omega \times I} : \Omega \times I \to Y$ is a pointed homotopy from the inclusion $(\Omega,y_0) \to (Y,y_0)$ to the constant pointed map.
Let $U$ be an open neighborhood of $y_0$ in $Y$. Since $\{y_0\} \times I \subset G^{-1}(U)$, we find an open neighborhood $V \subset \Omega$ of $y_0$ in $Y$ such that $V \times I \subset G^{-1}(U)$. Thus the restriction $G_{V,U} : V \times I \to U$ is a pointed homotopy from the inclusion $(V,y_0) \to (U,y_0)$ to the constant pointed map. This means that $Y$ is locally pointed contractible at $y_0$.
Observation 2.
Pointed $CW$-complexes $(Y,y_0)$ are well-pointed, but in general not first-countable at $y_0$.
Thus Griffith's theorem does not cover all relevant cases. We can only say that if $(Y,y_0)$ is well-pointed and first-countable at $y_0$, then the isomorphy of $\phi$ is a consequence of Griffith's theorem.
Observation 3.
There are spaces $(Y,y_0)$ which are not well-pointed, but satisfy (2).
An example is the two-dimensional Hawaiian earring $Y$ (see Union of Spheres is simply connected). This is simply connected and by "self-similarity" it is easy to see that it is locally simply connected at the basepoint $y_0 = (0,0,0)$. But $(Y,y_0)$ is not well-pointed because it is not locally pointed contractible at $y_0$.
The above obvservations show that conditions (1) and (2) are independent and the theorems based on them have a different scope.
Griffith's Theorem 3 says that
$\phi$ is always injective.
This allows to prove
Theorem. Let $(X,x_0)$ and $(Y,y_0)$ be based spaces. Suppose that $y_0$ has an open neighborhood $\Omega$ in $Y$ such that the inclusion $(\Omega,y_0) \to (Y,y_0)$ is pointed homotopic to the constant pointed map. Then $\phi$ is an isomorphism.
Note that the above assumption covers all well-pointed $(Y,y_0)$, but does not assume that $Y$ is first countable at $y_0$. Therefore the theorem generalizes the result based on (1), but is still independent from Griffith's result based on (2).
Proof of Theorem. It remains to show that $\phi$ is surjective.
Let $u : I \to (W,*) = (X,x_0) \vee (Y,y_0)$ be a loop based at $*$. The set $U = u^{-1}(W \setminus X)$ is open in $I$ and does not contain the boundary points $0,1$. The connected components of $U$ are pairwise disjoint open intervals $J_\alpha, \alpha \in A$. Let $A' = \{ \alpha \in A \mid u(J_\alpha) \cap (Y \setminus \Omega) \ne \emptyset \}$. We claim that $A'$ is finite.
Assume that $A'$ is infinite. Then there exists a sequence of pairwise distinct $\alpha_i \in A'$, $i \in \mathbb N$. Pick $s_i \in J_{\alpha_i}$ such that $u(s_i) \in Y \setminus \Omega$. Note that by construction each $J_\alpha$ with $\alpha \in A$ contains at most one $s_i$. The sequence $(s_i)$ has cluster point $\sigma \in I$. Since $u$ is continuous, $u(\sigma)$ is a cluster point of $(u(s_i))$. Since all $u(s_i)$ are contained in the closed set $Y \setminus \Omega$, we get $u(\sigma) \in Y \setminus \Omega \subset W \setminus X$.
Thus $\sigma \in U$. Let $\alpha^* \in A$ be the index such that $\sigma \in J_{\alpha^*}$. Since $J_{\alpha^*}$ is an open neighborhood of $\sigma$, infinitely many $s_i$ must be contained in $J_{\alpha^*}$. This is a contradiction.
So let $J_i = (a_i,b_i)$, $i=1,\ldots, n$, be the finitely many components of $U$ with image under $u$ intersecting $Y \setminus \Omega$. We have $u(a_i) = u(b_i) = *$. Note that the $K_i = [a_i,b_i]$ can intersect only in boundary points. Let $G : \Omega \times I \to Y$ be a pointed homotopy deforming $\Omega$ into $y_0$. We can extend it by the stationary homotopy on $X$ to a homotopy $G' : (X \cup \Omega) \times I \to W$. Let $I' = I \setminus \bigcup_{i=1}^n J_i$. Then $u(I') \subset X \cup \Omega$. Define
$$H' : I' \times I \to W, H'(s,t) = G'(u(s),t) .$$
We can extend it by the stationary homotopy on $K = \bigcup_{i=1}^n K_i$ to a homotopy $H : I \times I \to W$.
By construction $u$ is homotopic rel. $\{0,1\}$ to $v : I \to W, v(s) = H(s,1)$.
The set $I'$ is the finite union of closed intervals $J'_j = [a'_j,b'_j]$, $j = 1,\ldots, m$, which may be degenerate (i.e. $a'_j = b'_j$). Omitting such degenerate $J'_j$, we obtain a finite collection of closed subintervals $L_k = [c_k, d_k] \subset I$, $k=1,\ldots,r$ which cover $I$ and either have the form $J_i$ or $J'_j$. Two such $L_k$ can intersect only in common boudary points. All boundary points of the $L_k$ are mapped by $v$ to $*$. W.l.o.g. we may assume that $c_1 = 0, d_k = c_{k+1}, d_r = 1$.
The restrictions $v_k = v \mid_{L_k}$ are loops based at $*$ with images either in $X$ or in $Y$. Thus
$$[u] = [v] = [v_1]\ldots [v_r]$$
which is an element of $\pi_1(X,x_0) * \pi_1(Y,y_0)$.
