I would assume that, if the US subsidiary decided to comply, but the UK parent did not want them to, then the UK parent could say "no thanks" and there's nothing the US subsidiary could do.
To punish the parent, the US government could presumably prosecute the US subsidiary for noncompliance and, if successful, shut down all the company's US parts (including US-based domain names like .com) and seize US bank accounts etc.
And maybe even brand them as a terrorist organization since, clearly, if the US government says it's a matter of terrorism, if you don't help them then you're a terrorist (never mind that doing so would involve breaking the law of the UK jurisdiction where you're based).
And maybe even brand them as a terrorist organization
They are saving that for ideological enemies, such as people who hold traditional American political values (e.g. freedom of speech, freedom from excessive taxation).
To punish the parent, the US government could presumably prosecute the US subsidiary for noncompliance and, if successful, shut down all the company's US parts (including US-based domain names like .com) and seize US bank accounts etc.
And maybe even brand them as a terrorist organization since, clearly, if the US government says it's a matter of terrorism, if you don't help them then you're a terrorist (never mind that doing so would involve breaking the law of the UK jurisdiction where you're based).