You're not alone in having had challenges finding work in the UK. I've been here since May 2017, and all I've gotten was one one month part-time retail gig. I've applied to a lot of opportunities, from full time jobs in my field all the way to fixed term retail or festival jobs. I've tried to get a few part-time voluntary roles, but I haven't been eligible as they wanted UK referees. Right now I'm finally in the final round for a FT professional job I'm particularly well qualified for- I'll find out in the next few days if I got it!
My partner (Canadian) got a couple of brief summer temporary jobs when we first arrived and finally got a permanent, professional offer one year after we arrived. However, we had to relocate to a much higher COL area for him to take it and are lucky we could move.
My partner and I both registered as self-employed and have/had freelance clients, which is how we survived for so long without permanent jobs. It'd be more ideal if I had a regular job, especially in the higher COL area, but freelancing at least brings in some income, keeps my CV current, and fulfills the requirement to work my particular visa has. Any chance you have connections with companies back in the US that might contract with you remotely for some freelance projects?