First, let me fix generators for $H^*(SU(n))$ and $H^*(SU(n)/\mathbb T)$: For the first, consider the vector bundle on $\Sigma SU(n)$ with clutching map $\operatorname{id}_{SU(n)}$, i.e. with classifying map $f_n\colon\Sigma SU(n)\simeq \Sigma\Omega BSU(n)\to BSU(n)$, and let $\Sigma x_{2i-1} = f^*c_i$. For the second, let $\pi_k\colon \mathbb T\subset U(1)^k\to U(1)$ be the projection to the $k$-th coordinate and $L_k = SU(n)\times_{T,\pi_k} \mathbb C$ be the associated line bundle, and set $y_k = c_1(L_k)$; since $\bigoplus_{k=1}^n \pi_k$ is the restriction of the defining representation of $SU(n)$ to $\mathbb T$, the sum $E := \bigoplus_{k=1}^n L_k$ is trivial, so that all symmetric polynomials in the $y_k$ vanish since they can be expressed via Chern classes of $E$. The induced map $Q[y_1,\dots,y_n]/Q[y_1,\dots,y_n]^{S_n}\to H^*(SU(n)/\mathbb T)$ is an isomorphism; a basis for this cokernel is given by the monomials $y_1^{\alpha_1}\dots y_n^{\alpha_n}$ with $0\le \alpha_k < k$.
Now think of $p:G/\mathbb T\times\mathbb T\to SU(n)$ as an automorphism $\phi$ of $E\boxtimes\underline{\mathbb C}$; it splits as $\phi = \phi_1\oplus\dots\oplus \phi_n$, where $\phi_k = \operatorname{id}_{L_k}\boxtimes \pi_k$ is the automorphism of $L_k\boxtimes \underline{\mathbb C}$ obtained as the external tensor product of the identity of $L_k$ with the projection $\pi_k:\mathbb T\subset U(1)^k\to U(1)$.
Consider the clutching construction $(E\boxtimes\underline{\mathbb C})^\phi\to G/\mathbb T\times\mathbb T\times S^1$ which is obtained by gluing the two ends of $E\boxtimes\underline{\mathbb C}$ together using $\phi$. Denote by $z_k = \pi_k^*([U(1)]), u = [S^1]$ the obvious cohomology classes in degree $1$, and itdentify them and the $y_k$ with their image in the product. An easy argument shows that $c_1(\mathbb C^{\pi_k}) = uz_k\in H^2(\mathbb T\times S^1)$. By naturality, we have
\begin{align*}
(E\boxtimes\underline{\mathbb C})^\phi &\cong \bigoplus_{k=1}^n(L_k\boxtimes\underline{\mathbb C})^{\phi_k}\\
c_1\big((L_k\boxtimes\underline{\mathbb C})^{\phi_k}\big) &= c_1(L_k) + c_1(\underline{\mathbb C}^{\pi_k}) = y_k + uz_k\\
c\big((E\boxtimes\underline{\mathbb C})^\phi\big) &= \prod_{k=1}^n c\big((L_k\boxtimes\underline{\mathbb C})^{\phi_k}\big)\\
&= \prod_{k=1}^n (1 + y_k + uz_k)\\
&= \underbrace{c(E)}_{= 0} + u\sum_{k=1}^n z_k \sum_{S\subset \{1,\dots,n\}\smallsetminus\{k\}}\prod_{l\in S} y_l
\end{align*}
It is no surprise that this expression vanishes for $u=0$ since we can use the chosen trivialization of $E\boxtimes \underline{\mathbb C}$ to descend $(E\boxtimes \underline{\mathbb C})^\phi$ to $\Sigma(G/\mathbb T\times \mathbb T)$, and the resulting vector bundle has classifying map $f_n\circ \Sigma p$. Chasing through the definitions, we see that
$$
p^* x_{2i-1} = \sum_{k=1}^n\Big[\sum_{\substack{S\subset \{1,\dots,n\}\smallsetminus\{k\}\\|S| = i - 1}}\prod_{l\in S} y_l\Big]\otimes z_k\ .
$$
