Hi Irina and welcome to the UK!
It's a shame I don't live in London or I would be taking you up on the offer of lifedrawing or a coffee!
I'm afraid I won't be much help here as I'm not freelance, but maybe look into joining the AOI or other associations like that as they offer great advice and guidance with everything to do with illustration (invoicing, working with clients, putting together the best portfolio etc) it is pricey to join but probably worth it in the long run.
If you feel as though your portfolio is ready to be sent out to get some freelance work then I'd start researching some companies that you think would benefit from your work - advertising companies, web design, graphic design houses, galleries - that you think you could work with, then send them your portfolio!
My friend started her career of a graphic designer by finding local businesses that she thought could do with a new logo/shop front design/website etc and she mocked it up for free to show them, then they bought it off her and she immediately had something to put in her portfolio to show the next client. So maybe you could do something similar? Or even email your local council as sometimes they're looking for artists for advertisement material - especially libraries!
I would also get on social media and follow all the agencies/publishers/book shops/artists from those agencies that you like and maybe tag them in your work to get their attention or simply ask an artist to get their advice?
I'm sorry I'm no help with the accounting side of things, a lot of artists I follow on social media are freelance and I don't think many of them have accountants, they just learnt it all on their own so maybe you could save a bit of money doing it that way, or once things start moving for you maybe get some quotes for an accountants service and go from there.
The HMRC here is very confusing, I tried to declare myself as self employed a few years ago when I was freelancing but it all got a bit messy and I gave up and got a job lol! But I think with some research its easily doable, there might be some information on the Citizen's Advice Bureau website.
Let us know how you're getting on, sorry for the rambling message, it probably wasn't any help at all!