I think there is a possible reduction of this question. Consider a generic smooth torus embedding $f: T^2 \to S^1\times S^4$, then the projection to $S^1$ gives a Morse function on $T^2$. Because of the ambient dimension (codimension 3), I think one may isotope $f$ to cancel critical points and obtain an isotopic embedding $f': T^2\to S^1\times S^4$ such that the projection to $S^1$ has no critical points when restricted to $T^2$ if the map $f_{|T^2}:T^2\to S^1$ is homotopically non-trivial. Thus $f'^{-1}(z)$ is a union of circles in $S^4$ for all $z\in S^1$. The homotopy type of $f'$ should be determined by the number of circles, or correspondingly the index of the image $\pi_1(T^2)\to \pi_1(S^1)$. 

If the configuration space of $k$ circles in $S^4$ has $\pi_1$ isomorphic to the symmetric group $S_k$, then I think that one should be able to show that any two such maps are isotopic. 

For $k=1$, ie the case that $f'^{-1}(z)$ has a single circle, I think one can show that all such tori are isotopic. Because this is codimension $3$ in $S^4$ it is unknotted. 


Let $f: S^1 \times S^1 \to S^4$ be a map so that $f(z,S^1)$ is embedded for all $z$, then we may think of $f$ as a loop in in the configuration space of embeddings of a circle in $S^4$. This loop may be achieved by a 1-parameter family of diffeomorphisms by ambient isotopy. Let $D$ be an embedded disk bounding $\gamma= f(1,S^1)$, and let $D'$ be the disk obtained after the ambient isotopy. By the [4-dimensional light-bulb theorem (Theorem 1.10)][1],  $D$ is isotopic to $D'$ rel $\gamma$. 
So this isotopy may be achieved by a 1-parameter isotopy of disks. But any such isotopy is homotopic to the constant isotopy. Consider a point on the disk, the point gives a loop in $S^4$ which is contractible, and hence one may assume that the isotopy fixes this point. Then one may also assume that the tangent space to the disk at the point is fixed, and then that the disks are fixed by shrinking them down to the tangent space. Hence I think that the tori with $\pi_1(T^2)\to \pi_1(S^1\times S^4)$ surjective should be isotopic to the product torus.  




  [1]: https://www.ams.org/journals/jams/2020-33-03/S0894-0347-2020-00920-1/