To answer the question in the title:
It's not awesome at all.
I'm an English student at a Scottish university. I am effectively blind to my future as I have no idea what is going to happen to either of those countries.
As much as I feel your situation I think it's important to note a few things:
1) Nothing has changed right now. Britain is still in the European Union with all the rights and costs attached and will be until the 2 year period is up after Article 50 is triggered. From a timeline perspective you have at the very least 2 years before anything happens in regards to Britain's status with the EU, but more likely at least 3-5 years which should cover, or mostly cover, your studies.
2a) The language used by the SNP currently is a lot less impassioned about IndyRef 2. The language has shifted notably from pre-Referendum there absolutely will be an IndyRef 2 to "We're looking at all options and will discuss with Westminster" while keeping IndyRef on the table. The issue is there has now been 2 large referendums that Scotland has been involved with within the space of a couple of years, and it's likely Sturgeon is playing her cards close as there might be a lot less interest in having a third one so soon, and one on the same subject as a recent one. IndyRef1 was divisive enough, EURef more so. How would a closely contested IndyRef2 plan out is anyone's guess.
2b) IndyRef absolutely cannot happen in any clean manner without Westminster support. It's how the first one was able to run, and as much as Sturgeon bangs on that it'll happen, the very Act that allowed the creation of the Scottish Parliament contains an important Clause that gives Westminster the ultimate veto over anything created. If Scotland wants a managed independence and get the best gains from it if it should be successful, it absolutely needs Westminster's blessing and support and willingness to engage diplomatically. Right now the mood for that just isn't there, and it's likely to be something kicked into the long grass.
3) With all the above taken into account, none of the big players in UK politics fancies having a nasty scrap and are mostly taking a deep, collective breath and seeing how things pan out/negotiating. How long this honeymoon period lasts is anyone's guess, but right now it'd be political suicide to engage in any tit-for-tat measures to try and achieve aims. It's possible but I really strongly doubt the SNP will do something stupid like kick out all English Students from Scottish Universities, or cut their funding, nor do I think Westminster would try anything dodgy like pull money out of Scotland when May is scrambling to piece together a new National unity. Concilliation seems to be the message of the day.
So, in short, while yes the future looks scary it seems the politicians are atleast right now talking about 'options' and 'working together' rather then throwing demands and even if an IndyRef2 happened tomorrow it's likely negotiations on it would take a couple of years at the very least and during that time your rights and current priviledges should be protected.
Honestly, I think you should be worrying more about the possibility of your particular studies being dropped funding due to austerity measures of some kind, or some other bull, then anything drastic like being kicked out over politics. As for long term prospects within Scotland or further continuation of education in the coming 5-10 years I can't really predict, no one can, but considering most of Scotland's trade happens internally within the UK I highly doubt anyone with brain capacity is going to shut down the border. These are uncertain times, but nothing is going to change rapidly so long as everyone seems to be in a talking mood.
But that's just my two cents, maybe all the politicians will say fuck it and start a mutually destructive point scoring war with legislation.