Worth noting that not all parts of the UK are anti-EU - there is a larger margin in favour of the EU in Scotland than there is against in England:
http://newsnetscotland.com/index.php/scottish-opinion/6761-s...
[Edit: I am a Scot]
http://newsnetscotland.com/index.php/scottish-opinion/6761-s...
[Edit: I am a Scot]
This is IMHO highly unlikely in practice.
Succession/secession law regarding EU membership is untested, but the simplest suggestion is that rest-of-the-UK retains membership, and Scotland has to reapply as a "new" member. I think that's what you're saying. But, this is awkward and probably not legally correct because it treats the two newly formed countries differently.
This is also not in anyone's interest, except for perhaps Spain (Spanish politicians are pushing this solution because they are worried about Catalonia): it forces two countries to do the opposite of what their electorate would prefer.
For now, this is just a propaganda designed to frighten voters: some other arrangement would be made.
Wikipedia has a large amount of detail: https://en.wikipedia.org/wiki/Future_enlargement_of_the_Euro...
Given that much of England don't want to be in the EU perhaps the easiest thing would be that we adopt Spain's ideal approach but instead of England, Wales and NI remaining in and Scotland reapply, Scotland get the "existing" place and England, Wales and NI can reapply if they wish...
Edit: Englishman living in Scotland (for 7 years now with no intention of leaving whatever happens with independence).
The two most recent polls I'm aware of show a 3 point and 15 point lead respectively (both for rejecting independence) The low one comes from a polling company consistently showing the smallest leads, the high one from a company showing the largest leads. Looking at all polls somewhere in high single figures with wide error bars would probably be the safest assumption.
I'm not saying it won't happen but the suggestion that it's a small margin suggests it's more likely than it really is.
Slightly out of date (to end last month) polling data here: http://ukpollingreport.co.uk/scottish-independence-referendu...