If the UK split into two or more nations, would the resulting nations automatically be part of the EU?

I can certainly envisage England leaving the EU, or not applying for (re-)admission.  A lot of people outside England might see this as a good thing for the EU.

Also, the idea of an independent, self-governing Northern Ireland is quite worrying - given the total inability of the political forces there to work together so far.  Necessity might be the mother of invention, to take a very optimistic view.