The problem is that the museum field is saturated with job-seekers. By and large, working in a museum requires specialized knowledge beyond what you will get from a history degree: these days, you need to get a degree or at least certificate in museum studies to gain skills in label writing, handling and storing objects, cataloguing, mountmaking, etc. The major exception is for curators, but to make the jump from academic history training to museums there you typically need a PhD.

I am in the US, but my understanding is that the situation in the UK is even more dire, with fewer positions, lower salaries, and a more significant amount of free labor to get your foot in the door expected.

I have a B.A. in History and an M.A. in Museum Admin. I currently work in publishing and I love it. I get to interact with history every day and I get a hands on experience with business management and editing. My area isn't specific so I'm always doing research, fact checking, and writing on different subject matters. Don't limit yourself to museums, universities, or "history jobs", there is plenty of work out there that needs your skills sets and knowledge base. Competent editors are always in demand, as well as copy writers for text books, websites, newspapers, and government-run historic sites.

If you are truly committed to working in a direct history field, I cannot overstate how important it is to volunteer. Everyone I know who works in museums started by volunteering their time and knowledge with their organization. Every little bit helps and even if a job doesn't open up immediately, you will make valuable connections in the industry and a strong recommendation of work ethic and training before you even apply for a job. As others have said, hard work, moxy, and a good attitude goes a long way and volunteering is a great way to prove all three.