I'm a US citizen, lived my entire life in the United Kingdom.
If I were to earn capital gains from US assets bought and sold through a US bank account, do I have to declare those gains in the UK?
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As a UK resident taxpayer you're certainly potentially liable to pay capital gains on assets wherever they are in the world. See "Overseas assets" halfway down this gov.uk page.
Whether you have to declare them and/or actually pay any tax depends on whether the amounts and gains involved are above certain thresholds (around £11000); for more detail see the guidance accompanying Self Assessment and/or get advice. If you end up being taxed twice (because you've already paid some tax on the gain in the USA?) it's possible to get relief; see notes on CGT lower down this page.