I have a friend in television; he works as a second assistant director on Being Human and a number of other shows/movies.
The BBC might make a lot of shows and they might be a sponsor, but they don't "hire" people anymore for this sort of work. They haven't for years.
If you are looking at the production end of things then they hire on a contractual basis only. That means that when your show ends, you are out of work until you find something else and I don't think that will work for a work permit.
I work in television as an editor, mostly in broadcast news & sports, and what bookgrl describes is pretty common. Many production and post-production people are freelance/self-employed and don't work as permanent employees of a company.
There are staff jobs in corporate video, but you'd have a hard time getting one after just finishing university (or a master's degree). Most places would want you to have a few years of experience and a demo reel/website to show them before they'll hire you.
If I were in your shoes and really wanted to move to the UK, I'd look at the university route, and then work really hard at making contacts in the industry while in school in an effort to find someone at a company who might give you a shot.
That said, I don't know that even if someone wanted to hire you they could do so, given what everyone else has said about Tier 2 sponsorship requirements. Sorry I don't have anything more encouraging to say; the UK government is really trying hard to look tough on immigration, making life difficult for anyone who wants to come here.