Swinging in to answer this super late, I hope its still helpful!
I spent ages reading up on taxation laws in the US and UK to figure out how to be self employed. I eventually gave up and paid an accountant and it's definitely worth it. The US and UK tax small businesses different (the US as a person, the UK as a business) so tax law hits a whole new layer of complication when doing this. That said, it can be worth it depending on the business.
I picked up web design while on furlough and started my own business last year. Because of tax laws, I'm a sole trader/sole proprietor for now because it saves me a couple hundred quid a year on my accountants. My accountants have been really great for advice and do both US and UK taxes and seem really knowledgeable about how the laws interact with each other. It hurt really bad to pay them £870 to let me know I owe $16 in US taxes, but luckily business is already going well enough that by the end of 2021 I'll already be earning more than I was at any of the other UK jobs I worked or applied for in my area. So definitely worth it to me.
If you want to freelance, I'd plan on budgeting £1500-2000 a year for an accountant and decide if it's worth it based on that. I'm happy to share the names of my accountants or answer any other questions I can about the process