It gets worse. Your child, as a US citizen, is subject to something called PFIC (Private Foreign Investment Company) which limits opportunity to buy foreign investment trusts, child ISAs, certain pension arrangements not covered by the US-UK Tax Treaty article 18, bank accounts (subject to FBAR and its replacement FinCEN114, sometimes mortgages and credit cards (if they have a credit-balance option). The US tax obligations, should the child be an "accidental American" with no other connection with the USA other than place of birth (think: Boris Johnson) and the draconian penalties for not declaring assets and interests and trust funds (IRS forms 3520, 5471, 8938 and many, many more) are lead to severe anguish and costs: just the compliance cost in hiring a dually-competent accountant can be overwhelming. I have seen parents pay £500 to deal which a Child Trust Fund for one year, more than the £250 value of the fund.
Many of these rules, some of which started in the 1970s were not seriously enforced. But now, even failure to file a zero-income tax return to check the box at the bottom of Form 1040 Sched. B ("foreign account") and to file FinCEN114 when the total value of foreign accounts exceeds $10,000 at any point in the year even if interest income is insignificant, can be devastating -- although in fact this only happens when the IRS chooses to make an example. Or when the American citizen files for OVDP to regularise past noncompliance.
Much of this has little to do with lower-middle-class people who live abroad and have no assets, income or heirs in the USA. The filing obligation still exists: it relates to the US citizenship of the child (an American parent with a noncitizen child -- usually because the parent (perhaps your child 20 years from now) never lived in the USA -- has no PFIC restriction although the trustee relationship might be reportable on form 3510.
This is a new age. The UK Government, with its IGA compliance with FATCA, has given away the store. While it is unlikely that the anticipated mutual tax collection provision of tax treaties or the extradition provision for tax crimes will apply to British citizens who had that citizenship when the tax claim arose will be at risk so long as they are in the UK, we can't predict how this will affect dual nationals once they are outside the UK.
The US law on "Expatriation to Avoid Tax" includes a reasonable provision exempting dual nationals who renounce US citizenship at majority (but hey, that costs $2,350 now). Accidental Americans are collateral damage of laws enacted in response to Forbes Magazine's articles in the 1990s about wealthy Americans who had expatriated themselves, typically to Caribbean nations, to avoid tax.
American citizenship can be a great opportunity. But tax compliance requires a good deal of sophistication. Many can do it online with TurboTax -- if there are no PFIC complications (PFIC is, arguably, an illegal trade protection law disguised as a tax-equalisation measure). Basically the US -- with an old Common-law approach that considers "citizenship" as "allegiance" (Google: <Piggott Ligeance of the King>) and demands primacy of allegiance even from accidental dual nationals who have never set foot in the USA. Complying with PFIC and buying American equivalents (a 529 educational fund instead of a Child ISA for example) will lead to taxation in the UK of what is tax-free in the USA. And many tax-sparing arrangements will be doubly taxed with no relief from the US-UK Tax Treaty.
Sorry.
