That's a good question. it depends how long you wait and how strongly coupled they are.
For electrons and nuclear motion they are often strongly coupled enough that they quickly reach equilibrium and thus are at the same temperature (e.g. in liquids or solids). But on short timescales they can absolutely be at distinct temperatures. In dilute gases they can also become somewhat decoupled (there can be a well defined electronic temperature and nuclear temperature).
One case whether this can take a much longer period of time is when dealing with nuclear spin -- this is much more weakly coupled to the electrons and so can take much longer to equilibriate.
For electrons and nuclear motion they are often strongly coupled enough that they quickly reach equilibrium and thus are at the same temperature (e.g. in liquids or solids). But on short timescales they can absolutely be at distinct temperatures. In dilute gases they can also become somewhat decoupled (there can be a well defined electronic temperature and nuclear temperature).
One case whether this can take a much longer period of time is when dealing with nuclear spin -- this is much more weakly coupled to the electrons and so can take much longer to equilibriate.