Seq1[t_] = -S[t] κρ ( ψ1 W1[t] +ψ2 W2[t]) - D[S[t],t]
Jeq1[t_] = S[t] κρ (ψ1 W1[t] +ψ2 W2[t]) - γ * J[t] - D[J[t],t] 
W1eq1[t_]=α η *J[t] + δ2 W2[t] - (ξ1+δ1)*W1[t]-D[W1[t],t] 
W2eq1[t_]=α (1-η) * J[t] + δ1 W1[t] - (ξ2+δ2)*W2[t]-D[W2[t],t]

(* We first remove S[t] from Seq1[t] using Jeq1[t].*)

Solve[Jeq1[t]==0,S[t]][[1]]

Seq2[t]=Together[Seq1[t]/.Solve[Jeq1[t]==0,S[t]][[1]]/.D[Solve[Jeq1[t]==0,S[t]],t][[1]]/.Solve[W1eq1[t]==0,W1'[t]][[1]]/.Solve[W2eq1[t]==0,W2'[t]][[1]]]*Denominator[Together[Seq1[t]/.Solve[Jeq1[t]==0,S[t]][[1]]/.D[Solve[Jeq1[t]==0,S[t]],t][[1]]/.Solve[W1eq1[t]==0,W1'[t]][[1]]]/.Solve[W2eq1[t]==0,W2'[t]][[1]]]

(* Next, we reparameterize our equations only in terms of the proposed identifiable parameter combinations *)

Combos={p1==κρ ψ1,p2==κρ ψ2}
Z=Simplify[Solve[Combos,{ κρ,ψ1, ψ2}]][[1]]

Seq3[t]=Factor[Seq2[t]/.Z]*κρ
W1eq2[t]=Factor[W1eq1[t]/.Z]
W2eq2[t]=Factor[W2eq1[t]/.Z]

(* In this case, we have three input-output equations. We just need to make Seq2, W1eq1, and W2eq1 monic. *)

Monos1=MonomialList[Seq3[t],{J[t], J'[t],J''[t],W1[t],W1'[t],W1''[t],W2'[t],W2''[t]}];
Coeffs1=Simplify[Monos1/Last[Monos1]/.{J[t]->1, J'[t]->1,J''[t]->1,W1[t]->1,W1'[t]->1,W2[t]->1,W2'[t]->1}];
Monos2=MonomialList[W1eq2[t],{J[t], J'[t],J''[t],W1[t],W1'[t],W1''[t],W2'[t],W2''[t]}];
Coeffs2=Simplify[Monos2/Last[Monos2]/.{J[t]->1, J'[t]->1,J''[t]->1,W1[t]->1,W1'[t]->1,W2[t]->1,W2'[t]->1}];
Monos3=MonomialList[W2eq2[t],{J[t], J'[t],J''[t],W1[t],W1'[t],W1''[t],W2'[t],W2''[t]}];
Coeffs3=Simplify[Monos3/Last[Monos3]/.{J[t]->1, J'[t]->1,J''[t]->1,W1[t]->1,W1'[t]->1,W2[t]->1,W2'[t]->1}];
Coeffs=Union[Coeffs1,Coeffs2,Coeffs3]

(* Coefficients from either input-ouput equation are identifiable parameter combinations. We now must test whether the map from the parameters to the coefficients is injective.*)

xCoeffs=Coeffs/.{γ->g,η->h,δ2-> d2, δ1->d1, ξ2->x2,ξ1->x1,p1->q1,p2->q2, α->a}

Solve[Coeffs-xCoeffs==0,{γ,η,δ2,δ1,ξ2,ξ1,p1,p2, α}]

(* Because the map from parameter space to the coefficients of the input-output equation is injective, these nine parameters are structurally identifiable.*)