Scotland will almost definitely hold an independence referendum ASAP, and it's extremely likely that they will drop out of the UK, especially if the EU tosses them a bone and invites them back in.
Northern Ireland is a big question mark. I think this referendum may have brought NI closer to want to become independent (or unify with Ireland), again, especially if the EU offers them to join back in.
And I know you didn't ask about that, but it might even be in Gibraltar's interest to opt out of the UK as well.
It heavily depends on how the UK will fare once it's out of the EU. Will its economy implode or will it get away with just a few bruises? If it's the latter, then the EU will fall apart, most likely.
This is especially the case if the EU approaches Brexit softly and concedes to allowing the UK many of its rights as an EU member but without the responsibilities.