Is there a way to obtain the original signal (stationary process) from its combination through filtering (matlab) and crosscorrelation?
I think that (at least in general) (and if I understand your question correctly) this is not possible. As an example, assume that $w_1(t)$ contains a harmonic component of period $t_0$. The signal $x(t)$ will not contain any trace of this component. As a result no amount of filtering $x(t)$ will be able to recover the original signal.