Yeah - what Margo said. The difference between the UK and US in those terms is slooooowwwww. I also underestimated and it took me a year from landing to actually starting, and my partner about 7 months. A lot of close calls but that was an extremely frustrating year. Even when I went to my second job the only reason it went as fast as it did (< 6 weeks) is because I had someone very high-up recommend me and this is not even CLOSE to my dream job but given circumstances at the time and London costs, here we are.
I found those jobs by typing in "museum jobs Swindon" into google. Indeed came up with those listings, but I would also see what is available on LinkedIn - a lot of other roles seem to have moved on to LI. Heritage is another word used here that isnt in the US. To find jobs I would use Professional Association websites first, then Linked in, then Indeed. University of Leicester has a Museums JobsBoard too. Here are some more professional associations:
http://www.londonmuseumsgroup.org/resources/Personally, I would make up two resumes - one for museum jobs that may be heavy on the museum jargon, and one for broader cultural management/administration jobs that shows your ability to manage a project, manage stakeholders, etc, regardless of if its in a museum or not. The NERC is hiring in Swindon and they are looking for folks with admin skills and experience; it may not be museums, but itll get your foot in the door and you can start building experience at least. You have a very narrow career focus to date - to give yourself the best chance of success you will want to broaden your search a bit by identifying those transferable skills. Also - I havent found UK/US resumes to be very different - I use the same one, but the UK one has UK spelling, information on my work status at the top in black italics, and I have only highlighted relevant work areas (oh, and added an interest and hobbies section which they seem to like here for some reason).
Finally - I can't say I have ever seen a job application requiring my NI number - they will only care about your work number as the NI can get sorted later (in fact, when I worked my first job here in the late 90s I took off the morning about a week in to go get it done). Its not like a SSN.
I know its really scary looking for a new job and its even more scary looking for a job in a place where you know no-one, how the rules work, etc. Itll be painful and no one wants to go through that. But you can't learn the ropes until you start applying and looking and getting documents set up, potential sources identified, etc. A job is almost like the last gateway that you are fully resident (ok, maybe a bank account is too - at least in this country).