I don’t see how one trustee could make the appointment without the other joining in. The default position is that all trustee powers must be exercised jointly (ie unanimously)
If there were some power allowing one trustee to act alone (which I haven’t seen before) it would still be “interesting” in the sense of interesting to the trust litigation team.
Section 36(1) Trustee Act 1925 provides for a situation where a trustee “refuses to act” in connection with the trust. However, it sounds as though it is simply that there is disagreement, rather than the other trustee “refusing to act”.
I would say it probably does fall outside of refusing to act and more into the realms of an inability to understand the role of a trustee or to accept that the assets are held in Trust and not in their own name.