It is not clear. The closest previous example would be the dissolution of USSR. In that case, Ukraine based nuclear weapons are moved to Russia under the condition that the five members of UN permanent security council provide nuclear protection for Ukraine. However, Russia is vastly more powerful than Ukraine where Britain and Scotland are much closer in size, economy and population.
Anyways, IF the separation really occurs, both Britain and Scotland may not have the economy necessary to support UK's current nuclear arsenal. (Maintaining nuclear weapons are an expensive effort) In that event, it is possible some of the nuclear stockpile will be destroyed under UN permanent security council supervision.
Of course, the nightmare scenario would be some of the weapons ended up missing due to poor maintenance.