Hi! So sorry to hear that you're having trouble landing a job -- but trust me, you are not the only one. I had a few months of constant rejections (or worse, no response) from everywhere I applied, including part-time ones that I was certainly overqualified for (I have a BA and MSc). About 2 weeks ago, I started a part-time position at Waterstones, which I'm really enjoying but is a definite downgrade in pay and responsibility from the job I left behind in the US.
Unfortunately I can't offer any recommendations or references, but I thought I'd throw a few tips out that were really helpful for me. First, make sure that you address ALL of the essential items listed in the person description, and the desirable ones too if you can. Once I started doing that, I got a lot more feedback. It can be SO annoying and time consuming (especially if you're applying for not very well paid / part-time work!!) but it does make a big difference.
Second, if you can find somewhere to volunteer, that will help get UK experience on your CV (which is crucial) and may help you start to develop a network. I started volunteering once a week at my local Citizens Advice Bureau and, once that was on my CV, I was offered interviews at the next 2 places I applied (one of which is my current job).
Don't lose hope and keep plugging away -- something WILL come through and it's normal for it to take a few months to land that first job.