In terms of passport/immigration control, probably nothing at all. The terms of independence and the relationship between Scotland and the remainder of the UK will still have to be worked out.
One possibility is that Scotland signs the Schengen Agreement in which case there would be passport controls between Scotland and England and N.I - and then you'd get partitioning of stations and airports into Schengen and non-Schengen areas which is highly unlikely given the amount of inter-dependence between these (currently) two parts of the UK.
Someone might like to comment on international rail-travel regulations but to be absolutely honest, I think if Scotland (or Wales, NI, or even England!) became independent there would be some kind of treaty between the nations giving freedom of movement in excess of what is currently provisioned by similar treaties like Schengen. At minimum I would expect something like the treaty that used to exist between the Nordic countries regarding freedom of movement without password/immigration controls.
But this really is too much speculation given that even if Scotland votes yes, and the Westminster gives the ok, and a whole bunch of other things, there'll still be a huge
amount of politics to work out before any formal data of independence is agreed.
And then there's Scotland's membership of the EU, UN, what happens to the Royal Family and is Scotland a proper republic versus having Queen Elizabeth II as nominal head of state (or King Charles III, or by the time the above have been worked out, King William the IV?)...
...the trains, I guess no change...
t.
Ian