Job opportunities in the UK for foreigners

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Regarding the choice of Uni, it doesn't really matter as vast majority of them, if not all, are great. Just make sure the course that you're going to study is accredited by the IMechE ( https://www.imeche.org ). You should be able to find that information on the Uni website, on the course page. It's not mandatory, but it ensures the degree is taught to the latest professional standard and that you will be able to gain a Chartered Engineer (CEng) status, which could also be useful for your future career progression.

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Visa application: It seems that you plan to study in the UK with the goal to stay longer. There is nothing wrong with that, however your intentions when applying for the visa should state that your goal is to study in the UK, not to use the student visa route to remain in the UK. I can't remember the formal way that the UK Home Office put it, but basically they want to know you have genuine goal of studying here and not simply using the student visa route to live in the UK, however that doesn't mean that you can't, it just shouldn't be the only reason for you to apply for such visa. On top of that, the UK Government have recently increased the minimum salary requirement - You must be paid £35,800 or more a year in order to qualify for the Tier 2 visa (or work visa). This isn't what the majority of the companies would offer a graduate. Graduate programmes often pay somewhere between £28k and £35k, but you do have the 2 year graduate visa option to get some work experience and then get the salary you need in order to qualify for the work visa, but you must understand that the graduate visa programme may be closed before you graduate (the previous Prime Minister has been considering scrapping the graduate visa option), potentially leaving you with no options but to either try to find a high paying job and if unsuccessful return back to Korea.

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Costs: I don't know your financial situation, but Masters degree fees are quite high for international students - about £29,300 a year in Bristol Uni, £32,500 in Manchester Uni and £29,110 for Sheffield Uni. Also remember that you'll have to pay for everything out of your own pocket - rent (which is also quite high, I'd say at least £500/month on average), bills, transportation, food, cloths, etc. Not only you must make sure you have enough money to sustain yourself before you find an engineering job after you graduate, you'll also have to prove the UK Home Office that you do have such funds and they come from genuine resources (such as salary and savings).

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Career: Having a such a narrow career focus (materials engineering) may limit your choices and introduce another level of challenge in finding a job after you graduate. Personally, I'd suggest not limiting yourself too much with this and going for any relevant mechanical engineering job at a known company with good reputation (that's almost any company that advertises on the Gradcracker). Once you've moved from student visa to a work visa and once you've gathered enough experience then you'll have much better options with regards to your long term career. Also, by that time you might easily change your mind and stay in the industry of our choice, for example automotive, energy, healthcare, infrastructure, etc. All industries need engineers these days, so unless you're really bad at making good CVs and performing at job interviews, you should not have any issues finding a job later on. Except for defence and aerospace companies, and sometimes for companies with headquarters in the USA, you'll have some trouble working there as you're not an English national, for security and for international trade compliance reasons.