I've seen that the universal covering of $S^1 \vee S^1 \vee S^2$ is infinitely many $S^2$'s chain together by line segments where the endpoints are identified. But when we are wedge two copies of $S^1$, I'm not sure how to make the two line segments distinguishable, yet still keeping the cover simply connected.
Any help will be appeciated!
Since the sphere is simply connected, the universal cover of $S^1\vee S^1\vee S^2$ is the universal cover of $S^1\vee S^1$ with a sphere at every intersection point. Since distinct paths in the universal cover of $S^1\vee S^1$ never intersect, the issue that the OP brings up about two paths coming together never occurs.