UK can only leave the EU in 2 years time so 2018. They have to spend min 2 years sorting out the terms and HAVE to follow EU laws until then. There is no way to guarantee anything but universities have said that even when UK leaves, they have no plans to make EU students pay international fees.
It is also generally done so that if you started the course on certain fees, they will not change for you. Only for the new people coming in. So in theory if you started this year with 9k fees a year, you will continue to pay 9k even if it changes to 20k per year.
As for how welcome you will be, depends on where you live and how obvious it is that you're not british. As you have probably seen, racism went way up after the vote has passed. There is no guarantee that it will get easier. But UK will not be able to deport you just because they left. At most, they will ask you to get a visa once UK has actually left EU.
But again, there is no way to tell for sure what will happen in 2 years time