CD on sphere and ellipsoids are very different in complexity if you want the exact nearest points (helps forwarding physics to the instant of contact) before the contact. That's because when you transform back to the unit sphere, you loose euclidian properties (right angles no more right).

But if you just want a yes/no response (interection test) or determine where a contact point is (if there is one), it's ez to transform the issue to unit sphere/triangle. The intersection property or the contact point are obviously equivalent on both sides of the transfo.

And that's pretty ez. It's the same as getting the point of the triangle nearest to the origin. In theory every algo for this issue is mathematically based on (not necessarilly implememented) the normal projection on the triangle plane. It can also be seen as a transfo into the barycentric coords of triangle (affine transfo).

Else, to determine the nearest points, there has been a rather complete thread about ellipsoids recently here. The nearest point is on the support plane (face), the edges or the vertices of the triangle. It's very ez to deal with ellipsoid/plane. But point/ellipsoid or line/ellipsoid are more complex.