Let's assume that the relevant pillar $t$ of your curve is currently (exclusively) calibrated using the reference instrument $f_0$ at market quote $q_0$. The instrument could be a swap, a forward rate agreement, tenor basis swap...
In what follows, I simplify somewhat in using scalar expressions; in practice you may see gradients / vector valued functions popping up.
Outline
The sensitivity of an instrument $F$ to a very small change in the quote $q_0$, say $\mathrm{d}q_0$, when measured through the effect of the quote on the zero rate $r_t$, is approximately:
$$
\begin{align}
\mathrm{d}F&=\frac{\partial F}{\partial r}\mathrm{d}r\\
&=\left.\frac{\partial F}{\partial r}\left(-\left.\frac{\partial f_0}{\partial q}\right/\frac{\partial f_0}{\partial r}\right)\mathrm{d}q_0\right|_{r=r_t,q=q_0}
\end{align}
$$
where the term in the bracket stems from a requirement on our calibration instrument, evaluated at the perfectly calibrated zero rate level $r_t$
$$
\begin{align}
\mathrm{d}f_0(q_0,r_t)\stackrel{!}{=}0&\stackrel{!}{=}\left.\frac{\partial f_0}{\partial q}\mathrm{d}q+\frac{\partial f_0}{\partial r}\mathrm{d}r\right|_{r=r_t,q=q_0}\\
\Rightarrow\quad\quad\quad\mathrm{d}r&=\left.-\left.\frac{\partial f_0}{\partial q}\right/\frac{\partial f_0}{\partial r}\mathrm{d}q\right|_{r=r_t,q=q_0}
\end{align}
$$
Let us intrudce another calibration instrument $f_1$ with corresponding quote $q_1$ that is supposed to replace $f_0$. Assume further, that both calibration instruments will yield the same zero rate $r_t$. Then, we should be able to simply replace $\mathrm{d}r$ from $f_0$ with that from $f_1$:
$$
\begin{align}
\mathrm{d}F&=\left.\frac{\partial F}{\partial r}\left(-\left.\frac{\partial f_0}{\partial q}\right/\frac{\partial f_1}{\partial r}\right)\mathrm{d}q_1\right|_{r=r_t,q=q_1}
\end{align}
$$
where we have used the already existing valuation curve, and we of course assume that the 'other' instrument is perfectly priced at par as well. Thus, we can use the 'other' instrument's sensitivities, as valued versus the already bootstrapped curve. I.e., you can get your sensis from your front office system or such.
Example
Assume a single curve world. Our reference instruments are are vanilla swaps with annual fix/float payments and annual forward rate agreements. The rates are in agreement, i.e. we can use both for bootstrapping and arrive at the same discount factors and continuously compounded zero rates:
tenor swap rate% forward rate% Discount factor zero %
1 1.0000 1.00000 0.99009901 0.995033
2 2.0000 3.03030 0.96097845 1.990165
3 3.0000 5.13455 0.91404629 2.995802
Let's focus on the third tenor. The simple swap bootrapper yields (with quote $c=3.00\%$)
$$
S(c,r_1,r_2,r_3)=c(e^{-r_1}+e^{-2r_2}+e^{-3r_3})-(1-e^{-3r_3})=0
$$
with sensitivities
$$
\begin{align}
\frac{\partial S}{\partial c} &= (e^{-r_1}+e^{-2r_2}+e^{-3r_3})=2.86512374824004\\
\frac{\partial S}{\partial r_3} &=-3D_3(1+c)=-2.82440302853815
\end{align}
$$
If we had used a simple forward rate agreement with quote $f=5.13455\%$
$$
F(f,r_2,r_3)=D_3\left(Fwd(2\to 3)-f\right)=D_2-D_3(1+f)=e^{-2r_2}-e^{-3r_3}(1+f)=0
$$
with sensitivities
$$
\begin{align}
\frac{\partial F}{\partial f} &= -D_3=-0.914046287552799\\
\frac{\partial F}{\partial r_3} &=3D_3(1+f)=2.88293535235877
\end{align}
$$
and thus
$$
\begin{align}
\mathrm{d}r&=1.01442\mathrm{d}c \quad\quad \mathrm{vs\ \ swap}\\
\mathrm{d}r&=0.31705\mathrm{d}f \quad\quad \mathrm{vs\ \ forward}
\end{align}
$$
A 1bp shift in the forward rate will shift the zero by 0.317 bp, whereas a 1bp shift in the swap rate will shift the zero by approx. 1bp as well.
