I graduated with an MLitt in History of Collecting and Collections from Glasgow Uni in 2010, back when the Tier 1 Post-Study Work visa still existed. Even with not needing a sponsor, I had a tough time finding a job.
However, I was limited to staying in Scotland because I was living with my boyfriend at the time, and museums/galleries are just tough to break into anyway. I'd been volunteering with one museum for over a year, in addition to volunteering at a NTS heritage site.
But, I also had trouble just finding any job NOT related to my field. It's really tough, but you might get lucky if you network and intern.
Also: I loved studying at Glasgow Uni and it was one of the best years of my life! Enjoy it!