frequency response of a signal in which two signal added in
What you ask for should rather be called a frequency spectrum or Fourier transform of the resulting signal. The term frequency response is reserved for linear time invariant systems.

Anyway, the answer to your question can be found using the z-transform of the signals involved. First let's expand the two source signals into a series of unit delays:

$$S_1(z) = \sum_{k\in\mathbb{Z}} a_k z^k$$
$$S_2(z) = \sum_{k\in\mathbb{Z}} b_k z^k$$

Your combined signal can be written as

$$F(z) = \sum_{k\in\mathbb{Z}} a_k z^{2k} + \sum_{k\in\mathbb{Z}} b_k z^{2k+1}$$

and we can plug in $z=\exp(i\omega)$ to get the Fourier transform:

$$F(\omega) = \sum_{k\in\mathbb{Z}} a_k \exp(ki2\omega)+\sum_{k\in\mathbb{Z}} b_k \exp(ki2\omega)\exp(i\omega)$$

Using the Fourier transforms from the z-transforms above we can see that this actually equals:

$$F(\omega) = S_1(2\omega) + S_2(2\omega)\exp(i\omega)$$

You can interpret this result as the additive combination of the two original signal spectra, each scaled in frequency direction by $1/2$ and one phase shifted proportionally to $\omega$.

Since your original signals are of finite length, your initial spectra are sinc-shaped, centered at the same frequency and with a constant phase shift. I'll leave working out the details to you.