How can i find this universal cover?

Question

I have $X = \{(x,y,z) | x^2 + y^2 = 1, z = 0\}$ and $Y = \{(x,y,z)| (x-1)^2 + y^2 + z^2 = 1\}$. What is the universal cover of $X \cup Y$?

Answer 1

$X \cup Y$ is sphere union two (kind of) diameters. So I think the universal cover is like:

Answer 2

@Wvk is correct in his answer, the universal cover looks exactly like the universal cover of the wedge of two circles pictured below,

except that every node of this graph has to be replaced by a sphere like so: