SNR After Multiplying Two Noisy Signals
The multiplication of the two noisy signal gives

$$(x_1+n_1)(x_2+n_2)=x_1x_2+x_1n_2+x_2n_1+n_1n_2=x+n\tag{1}$$

with the desired signal

$$x=x_1x_2\tag{1}$$

and the noise part

$$n=x_1n_2+x_2n_1+n_1n_2\tag{2}$$

Assuming all signals are independent of each other and have zero mean, we get for the signal power

$$\sigma^2_{x}=\sigma_{x_1}^2\sigma_{x_2}^2\tag{3}$$

and for the noise power

$$\sigma_n^2=\sigma_{x_1}^2\sigma^2_{n_2}+\sigma^2_{x_2}\sigma^2_{n_1}+\sigma^2_{n_1}\sigma^2_{n_2}\tag{4}$$

For the total SNR you get

$$\text{SNR}=\frac{\sigma_x^2}{\sigma_n^2}=\frac{\sigma_{x_1}^2\sigma_{x_2}^2}{\sigma_{x_1}^2\sigma^2_{n_2}+\sigma^2_{x_2}\sigma^2_{n_1}+\sigma^2_{n_1}\sigma^2_{n_2}}\tag{5}$$

With $\text{SNR}_1=\sigma_{x_1}^2/\sigma_{n_1}^2$ and $\text{SNR}_2=\sigma_{x_2}^2/\sigma_{n_2}^2$ this can be rewritten as

$$\text{SNR}=\frac{\text{SNR}_1\text{SNR}_2}{\text{SNR}_1+\text{SNR}_2+1}\tag{6}$$