As Cat says, a lot depends on whether your US student loans actually cover your living expenses at Imperial. Before you know that, though, you could do some tentative planning.
I've spent more than 100 nights in hostels in Europe, and I'm in my 70s, so you can do this. Hostels indeed are much, much cheaper than hotels in most cities, although if you have to have a private room, that's when you need to start comparing with hotels because the hostel won't necessarily be cheaper. Get on hostelworld.com for fast comparison of hostel costs in various cities and towns. Bus, train, and budget airline make getting around affordable in most countries, but I have to say the British Isles are the most expensive trains I've encountered by far.
As said already, the further east you go in mainland Europe, the more likely you will find affordable prices, although of course there are exceptions. Think about what you want to do in each potential town and then figure out how much those obvious sightseeing destinations will cost. (London is great in that respect, many free museums.) Some inexpensive locations to consider: for big cities Berlin is youthful and can be quite economical, Athens in off-season, Budapest although I haven't been there yet, Poland, smaller cities. Yet it's great if you can at least get a taste of the big three: London, Paris, Rome.
Browse the Watch, Read, Listen part of this website (choose Travel Articles under "Read"), also check out nomadicmatt.com. Some of his numbers will be dated, but still he'll be correct on order of magnitude. For hiking, think about what experiences you want and then look for economical destinations. The Alps are obvious for hiking, but you don't need to do the Swiss Alps--look at Austria, Italian Dolomites, etc., for example.
I don't know that any of us can tell you if your savings will be enough, but think and plan, then prioritize, and then just start traveling when there's an opportunity. Anything you can experience will be a bonus and probably give you clues about how to plan travels later in life.