The electron cloud for the single electron orbital $1s^1$ is radially symmetric in 3 dimensions. 
This is perhaps because of the probability distribution function (p.d.f.) $|\psi|^2 = ce^{-\frac{2r}{a_0}}$ depending on a single variable $r$. On the other hand, the p.d.f. of He atom is the bivariate function depending on the positions of both electrons $|\psi_{\text{He}}|^2 = ce^{-\frac{r_1 + r_2}{a_0}}$. Moreover, both electrons have opposite spins. Will this modify the distribution of dots in the electron cloud? If yes then how? More exactly, what is $\psi^2 (x,y,z)$ for each electron in the actual He atom (not the unperturbed He atom in which both electrons don't interact)? Do the electrons lose their individuality so that it is impossible to give separate wavefunctions for each interacting electron?
