Perhaps it would be helpful to start with some general comments.
A country in which an individual is resident generally has primary taxing rights. For cvc8445 this means the UK. Most countries also tax income that arises from sources within their countries, even for non-residents. Some tax capital gains, but usually only in respect of land related gains. The US, almost uniquely, has chosen to tax its citizens, irrespective of residence, on their worldwide income and gains.
Against this general background, the UK US double tax treaty sets out who has primary taxing rights and so forth. The treaty should be read with care. Most its provisions do not apply to UK resident US citizens. There is a reduced set of rules which do apply.
The provisions which do apply confirm that the UK can tax gains on the sales of US situated assets. (This is subject to special rules for land related gains.) It is then for the US to give credit for the UK tax.
(The double tax treaty provides for some situations where the UK must give a form of credit for US tax. In general, when US dividends are paid to a UK resident the UK resident can claim to reduce the normal 30% withholding tax to 15%. For UK resident US citizens, this is modified. The UK taxes the US dividend giving a 15% notional tax credit. The US then gives credit for the UK tax computed in this way.)
The reference to the tax return notes made by cvc8445 is misleading. Claims can only be made when the double tax treaty allows this. It is not a case of simply claiming any foreign tax.
It is too late for cvc84445 to amend the self-assessment for 2012. The time limit for amendment is broadly 12 months after the filing deadline.
There might an avenue to consider to saying to HMRC that all, or at least some, of the past returns where remittance basis was claimed were erroneous because the domicile of origin in the UK when cvc8445 returned to the UK in 2006, or at a date after this. This would need careful, detailed consideration.
I cannot further comment on the details of the US rules for giving credit for foreign tax. Perhaps others would confirm the mismatch, or make other suggestions.
In passing, the changes to non-doms from 6 April 2017 include specific rules for returning individuals who had UK domelike of origin. Broadly they deem the individual to be UK domiciled for tax purposes even where this would not be the case under international private law principles.