According to one English Vice-Chancellor, it would be the other way round, and Scottish Universities, if Scotland became indeoendent, would have to start charging fees to prevent being swamped by rUK students:
"Any attention on universities during the campaign has been on fees. Scottish Nationalist politicians insist that the status quo will remain - Scottish and EU students outside of the rest of the UK have their £1820 fees paid for by the government in Holyrood; English students on the other hand are charged up to £9000.
But Yes campaigners are misguided. This discriminatory approach seems unlikely to continue in an independent Scotland. Student mobility is enshrined in EU law, as is the right not to be discriminated against on the grounds of nationality. There is absolutely no precedent of an EU member state successfully applying for derogation from the provisions on student mobility.
So assuming an independent Scotland becomes a member of the EU, students from England, Wales and Northern Ireland will enjoy the same privileges as their Scottish peers, putting huge pressure on the Scottish higher education budget. Universities in Scotland receive around £150 million from rUK students, but that cost is likely to be much more in an independent Scotland with no tuition fees because Scottish universities would become extraordinarily attractive.
After all, why go to Durham or Newcastle and pay £9000 a year when you could go to St Andrews or Edinburgh and pay nothing? Scottish universities would be swamped with highly qualified applicants from rUK reducing available places for Scottish students.
The Scottish government would then have the choice of Scottish taxpayers subsiding English students or introducing tuition fees. It would surely be the height of irony if independence forced the Scottish government to copy, or come close to copying England’s higher education policy?"
The full article is at The Conversation