As is usual when it comes to taxation, it all depends upon the *whole* of the circumstances.
Have you looked at the links I've provided for you? They give the detail regarding domicile/residence and where it comes to it.. arising/remittance basis.
It's really difficult to give anything other than general pointers. There is no hard and fast 'rule' on it because it all depends upon the individual's situation.
There is no residency issue. Client does not enter UK soil even for vacation.
Since treaty say it is to be treated as business income is where I am lost. Businesses generally cannot cross geographic borders without both administrative and tax issues.
Further complicating thing US client is organized as a business under US law not just acting as a personal service provider. Thus it is UK business paying US business for services rendered.
If it was most other US tax treaties, I would say the US client made a mistake creating a US business or at least for the UK business not use the US business; rather for the UK client accept payment in her personal name. Thus she would meet treaty terms.
But again US-UK treaty seems to be quite different than most US tax treaties because it says such compensation should be considered business income. This would dictate, it does not matter if the US client is operating as a US person or as a US entity. It is still business income. And that again brings a host of administrative and tax issues.
I guess I have to delve to other sources.