This is a follow-up to my previous question about chern numbers. Write $\mathbb{P}^n:=\mathbb{P}^n_\Bbbk$ for projective space over some field $\Bbbk$ and assume that $X\subseteq\mathbb{P}^n$ is a linear subvariety, $\mathbb{P}^m\cong X$, say. I now consider the blow-up of $Y:=\mathbb{P}^n$ in $X$, yielding a blow-up diagram $\begin{matrix} \tilde{X} & \xrightarrow{\; j\;} & \tilde{Y} \\ \hphantom{\scriptstyle g}\downarrow {\scriptstyle g} && \hphantom{\scriptstyle f}\downarrow {\scriptstyle f} \\ X &\xrightarrow{\;i\;} & Y \end{matrix}$ My question is, what is the second chern class $c_2(\tilde Y):=c_2(\mathcal{T}_{\tilde{Y}})$ of the tangent sheaf of $\tilde{Y}$?
Remark: I am ultimately interested in the degree of $c_2(\tilde Y)c_1^{n-2}(\tilde Y)$.
My thoughts so far: As you can see from my first question, the chern classes of $Y$ (and $X$) have well-known representation, and there is a formula for computing the chern classes of blown-up varieties in Fulton's book Intersection Theory, namely Theorem 15.4. For brevity, I will quote his Example 15.4.3, which gives a formula for $c_2$:
$c_2(\tilde Y) = f^\ast(c_2(Y)) - j_\ast\left( (d-1) g^\ast(c_1(X)) + \tfrac{d(d-3)}{2} \zeta + (d-2) g^\ast(c_1(\mathcal{N})) \right)$
Here, $\mathcal{N}=\mathcal{N}_{X/Y}$ is the normal bundle of $X$ in $Y$ and $\zeta$ denotes $c_1(\mathcal{O}_{\tilde{X}}(1))$.
Of course, $c_1(\mathcal{N})=c_1(i^\ast\mathcal{T}_Y)/c_1(\mathcal{T}_X)$ by the well-known exact sequence $0\to\mathcal{T}_X\to i^\ast\mathcal{T}_Y\to\mathcal{N}_{X/Y}\to 0,$ but I am not 100% sure what effect the pullbacks and push-forwards have. In the end, I would like to express $c_2(\tilde Y)$ as the sum of intersections of well-known divisors in $\tilde Y$, like the strict transform of a hyperplane and the exceptional divisor.