I am amazed that there are so many replies to a serious query that is so beyond the scope of Mumsnet. And unsurprised that most (perhaps all, I haven't read most of them) are not of great quality. Suffice to say I know something about the substance of "accidental Americans". You will find that there are groups of activists on Twitter and on Facebook. And that there are dozens, maybe many dozens, of accountants and lawyers eager, as the "compliance condors" they are, to take your money. There are two good American lawyers who specialise in renunciation and associated tax issues: John Richardson (search for him by name, as Toronto lawyer (he's a dual national)) and Phil Hodgen. John is the more reasonably priced -- far more reasonably than any of the dozens whose work I have had reason to judge. I am a retired tax and bankruptcy lawyer.
The first and most important advice is not to panic. Nothing is going to happen to you. If you have no assets, income or heirs in the USA the IRS is not going to pursue you. The issue you face (as you noted) is the possibility that a UK financial institution may refuse to do business with you until and unless you get a Social Security number (or ITIN, but you can't have an ITIN b/c you're a US citizen). Your assets cannot be seized for payment of U.S. taxes. Unfortunately a U.K. accountant or lawyer is stuck with a rather unfortunate U.K. "ethics" rule that requires him/her to insist you pay all your taxes, domestic and foreign. That's why I discourage you from contacting any UK-based professional, who anyway would have to charge you 20% VAT.
You can, as one of my (many) daughters did, renounce U.S. citizenship on your own. You have to download some forms from the U.S. Embassy's consular Web site and you have to make an appointment, answer some questions, pay $2,350 by credit card, and swear an oath. A few weeks later you get a Certificate of Loss of Nationality and your name will (probably) be published in the Federal Register.
A "compliance condor" would tell you that you must get a SSN, fill out 5 years' back tax returns and another one for the current year, complete IRS form 8854, and pay any back tax due. You also will be told you have to fill out FBAR forms (that is done online, Google "FBAR"). If you fail to do so, you become a "covered expat". The significance of that is that any American citizen or resident who inherits from you will have to pay 40% tax. You could also owe capital gains tax on the deemed value of all your assets and pensions except State Pension. If you do file form 8854 (which means you also need a SSN and would probably want to use the IRS Streamlined practice which is usually penalty-free (Google: site:irs.gov streamlined -- and watch out for private companies and shyster lawyers who want to frighten you and get big fees).
A lot depends on your age and your future intentions. It's unlikely you have minor children for whom you want green cards because they are not US citizens from birth (because, like my daughter, you never lived in the USA). The cynical but practical decision might be just to renounce for $2,350 and do nothing else. Once you have a CLN your bank will no longer want IRS form W-9 but maybe will want form W-8BEN, which is used by non-US and ex-US citizens.
If you have questions I'll be glad to answer either here on on FB (as Sora Fon). I am not your lawyer, I do not take clients, I cannot be paid.
One last reference: Google "FATCA Globalex" for a long and scary article on the penalties asserted against Americans who were actually guilty of hiding money abroad and not declaring it or paying tax on the dividends, interest and gains. That doesn't apply to you: you were not wilful, you are innocent, you knew nothing.
Best of luck.