I just accepted a contract to join an investment fund (small boutique) and have applied for a conversion from tier 4 to tier 2.
My 2 cents: I knew many people in the same position as me at the LSE (where I studied). I've seen people with Deutsche Bank internships not get sponsorships with First Class grades in Economics, Data Science Masters students with stellar grades and work experience not get a sponsorship either.
Its brutal out there. I did a lot of research into this in my first year and realized that my university priorities would be very different from my European flatmates, as the pressure to get awesome work experience was higher for me than them - they can look around for internships after graduation and then find their path - some people with firsts at even the top uni's do this. Only a handful walk into their ideal job right away (15% maybe, and a good portion of them work for consultancies or banks, when their first internship comes in some other country, in many cases through a connection).
In my second year I can't even explain the stress of holding a part time job to pay my bills, uni studies and then the relentless pressure to get a good set of relevant work experiences to have a chance of getting a tier 2 visa. Thing is a lot of the tier 2 jobs go to the big 4 account firms - pardon my language, but I didn't kill myself for 3 years to work in an accounting or audit job that is of no interest to me, to earn 28k pre-tax in central london. That's garbage. Employers know they have an over-supply of talent from all over Europe and a ton of good international students, and the wages barring the big 3 consultances, a few boutiqe finance firms and the Investment Banks are really not great. That's digressing a bit, however.
The only real way to get the sponsorship is actively check the tier 2 list of sponsors (just search for it on google, there's a pdf on the home office website with an updated list). Don't bother applying to any company not on there - a brand new sponsor license takes 3+ months and is very expensive. Unless you're a genius who is going to change the world, its not going to happen. It sucks when you see a job you love but can't apply for as the company can't sponsor you...you have to suck it up, i learned to.
I've seen people with nowhere near my skillset (pardon the self aggrandization) but I worked relevant part-time roles for my final 2 years and got stellar internships and grades, and built a hell of a network in the field of my interest - people with half these credentials walked into roles i wanted, but couldn't even apply for because I have an Indian passport. That's ridiculous, absolutely ridiculous. I've paid 54k in uni fees alone ffs. The rules are so demented its beyond belief.
End of the day, I felt sad many times, lost hope - my family took out a loan to send me here. No way I can pay them back with the pay in my home country in the field I'm interested in. Lots of bad stuff comes into your mind. Apply lots - know you'll have to work 10x as hard in the interview/job process as your european counterparts, but that it is possible. I applied to 20+ startups and 20+ investment firms who do the work I like. I got 2 offers rescinded because even though the companies do sponsor applicants...it still costs them a few thousand to sponsor someone from tier 4 to tier 2 (this was told to me by both firms so I think its legit. Days and days of preperation, doing assesments...and it ends with that. Eventually, this week, I got a great offer at one of my top choices.
The date of my first application for a graduate job was October 1, 2017. I already had 2 good summers of relevant experience and a full year of part time eexperience in my second year at an investment fund, and like an 81 average at one of the best uni's in the country. Took me 11 months, around 40-45 rejections, 2 rescinded offers to get here, a month before my student visa expired.
Moral of the story//TL;DR: Its a **** system. Most people not in your shoes won't understand it. Your second year and third year, or if a masters student, your entire masters year should focus on job search/experience and then degree - all you need is a 60% really. No one rejects you if you don't have a first. Build a network, hustle like crazy and suck up the rejections - and remember its only 3.5% of all internationals who get these visas. Don't feel sorry for yourself, become valuable enough as a brand and product yourself that someone will hire you on a good salary.