Someone working in the UK for a single US employer on a contract basis is quite likeky to be an employee under English or Scottish law - not self-employed. The US version of an independent contractor is not replicated here in the UK.
This does not mean one cannot be self-employed - but one would need a contract that lets the worker substitute other workers, set prices, have several contracts at once and similar "status" factors.
If one is an employee one can benefit from UK employment law such as paid vacatiion, sick pay, maternity pay and statutory redundancy.
On the downside deductibilty of expenses for tax purposes is more restrictive for an employee versus a self-employed worker.
There is a high risk to a US company who uses a UK based independent contractor because such a contractor is likely to constitute a permanent establishment of the US entity requiring filing and payment of UK corporate taxes by the US entity.
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